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In this article, the phenomenon of countercurrent imbibition in a specific displacement method occurs within a dipping homogeneous porous medium. This phenomenon plays a vital role in the process of oil recovery, which motivates our analysis. To overcome the difficulties associated with the nonlinear partial differential equation governing the countercurrent imbibition phenomenon, we have been using two distinct methods: the natural transform decomposition method and the variational iteration transform method with their convergence analysis for solutions. These techniques give different perspectives and enable us to obtain approximate solutions. Notably, both the proposed methods demonstrate the potential for substantial production of oil during the secondary oil recovery process in the petroleum industry. The obtained results by the proposed methods show the efficiency in optimizing the oil recovery rate. [ABSTRACT FROM AUTHOR]

The steady MHD boundary-layer axis-symmetric flow of a third-grade fluid passing through an exponentially expanded cylinder in the vicinity of a magnetic field is investigated in this study. The problem is mathematically modeled. Suitable similarity transformations are carried out to convert the partial differential equations into nonlinear ordinary differential equations. The Runge–Kutta fourth-order shooting technique is used to solve the transmuted system of nonlinear ODEs. Graphical representations of numerical findings are used to examine the effects of various physical factors on the velocity and temperature profiles. The influence of fluid variables on the velocity curve, such as third-grade parameters, second-grade parameters, and Reynolds number, is illustrated and explored. The skin-friction coefficient expression is computed and given. The widths of the velocity and momentum boundary layers are revealed to be increasing functions of the curvature parameter. It is found that the third-grade fluid has a higher velocity profile than Newtonian and second-grade fluids. Also, the stretched cylinders cause a more progressive shift in heat and mass pattern for flow than flat plates do. [ABSTRACT FROM AUTHOR]

This paper presents an improved Cham method as an efficient technique for obtaining analytical exact solutions to nonlinear partial differential equations. We apply this method to solve the nonlinear Schrödinger equation in conformable operator, a challenging equation frequently used in various scientific fields. The method enables us to derive different traveling wave solutions, such as kink, coindal waves, breather waves, periodic singular solutions, and periodic multi-wave solitons. We also provide graphical representations of some of the obtained solutions to help understand their dynamic characteristics. These results highlight the effectiveness and adaptability of the method and demonstrate its potential to solve other partial differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we study the Fokas–Lenells equation, which is a derivation form of the nonlinear Schrödinger equation and can be used to describe nonlinearity in the propagation of optical pulses. To seek solutions for this nonlinearity, the Logistic method has been proposed in order to drive acceptable and understandable results regarding physical meaning. This method is defined based on the summation of an ordinary differential equation. Moreover, for further study, the Bernoulli (F ′ / F) -expansion method is used by considering hypothetical solutions as a function of the Bernoulli equation. Subsequently, solutions of the Fokas–Lenells equation are successfully acquired, displaying efficient results in hyperbolic, trigonometric, and exponential equations. The importance of these results appears in the definition of the wave function under the consideration of the appropriate coefficients. Hence, computer simulation is used for better understanding of the generated results. The dynamic behaviors of these solutions are demonstrated in 3D graphs, contour maps, and line plots, under applying various parameters, solutions of the waves show different soliton behaviors, including bright-dark, bright, and breather solitons. The results indicate that the exerted methods are novel, reliable, and effective approaches which can be employed in a wide range of nonlinear differential equations. These methods and their begotten results are far from the complexities of mathematical structures and therefore go beyond previous efforts in the literature. [ABSTRACT FROM AUTHOR]

This paper delves into the intricacies of the (2 + 1)-dimensional asymmetric Nizhnik–Novikov–Veselov ( ℕ ℕ ) model, a nonlinear partial differential equation governing ion-acoustic wave propagation in plasma. By employing advanced analytical and numerical approaches, the study explores innovative solitary wave solutions, particularly focusing on the dynamics of isochoric flow. Isochoric flow analysis is crucial for unraveling the complex behaviors exhibited by incompressible fluids like elastomers and bio-elastomers, which maintain a constant density. The derivation of the (2 + 1)-dimensional ℕ ℕ equation stems from fluid equations governing plasma dynamics. This model serves as a valuable tool for simulating experimental observations of plasma waves. The computational methodology applied in this research demonstrates a commendable level of precision and consistency, yielding novel solitary wave solutions previously unreported in the ℕ ℕ model. These results underscore the study's importance and novelty. The outcomes not only contribute to our understanding of incompressible fluid dynamics, but also lay the groundwork for future investigations in this domain. The revealed solitary wave solutions have the potential to inform the development of more accurate models for predicting fluid dynamics, thereby advancing the field. [ABSTRACT FROM AUTHOR]

We argue that the spatial discretization of the strongly nonlinear Lefever–Lejeune partial differential equation defines a nonlinear lattice that is physically relevant in the context of the nonlinear physics of ecosystems, modelling the dynamics of vegetation densities in dry lands. We study the system in the lattice ℤ2$$ {\mathrm{\mathbb{Z}}}^2 $$, which is especially relevant because of its natural dimension for the emergence of pattern formation. Theoretical results identify parametric regimes for the system that distinguish between extinction and potential convergence to nontrivial states. Importantly, we analytically identify conditions for Turing instability, detecting thresholds on the discretization parameter for the manifestation of this mechanism. Numerical simulations reveal the sharpness of the analytical conditions for instability and illustrate the rich potential for pattern formation even in the strongly discrete regime, emphasizing the importance of the interplay between higher dimensionality and discreteness. [ABSTRACT FROM AUTHOR]

Image denoising is one of the fundamental problems in image processing. Convolutional neural network (CNN) based denoising approaches have achieved better performance than traditional methods, such as STROLLR and BM3D. However, CNNs can easily bring unexplainable artifacts to denoised images. In this article, a Deep Learning‐Informed Diffusion Equation (DLI‐DE) framework utilizing the image prior or the image gradient prior for image denoising is proposed. The image priors and gradient priors are learned from CNN models and used as coefficients in diffusion equations. The solution of DLI‐DE is infinitely smooth from the uniqueness of existence theorem, which guarantees that the denoised image is free of artifacts. Good properties of DLI‐DE also ensure high‐quality of denoising. The experimental analysis confirms that the denoising performance of DLI‐DE is comparable to that of contemporary CNN‐based denoising methods such as TNRD and DnCNN, while effectively preventing artifacts. [ABSTRACT FROM AUTHOR]

In this paper, the similarity solutions for the shock wave propagation in a perfectly conducting dusty gas under the effect of axial or azimuthal magnetic field in the case of rotating medium by using Lie symmetry method have been discussed. The optimal system of one-dimensional Lie subalgebras related to the hyperbolic system of nonlinear partial differential equations had been constructed by using general invariant function and general adjoint transformation matrix. The optimal system is the collection of inequivalent optimal classes, which provides crucial information regarding different types of the similarity solutions. On the basis of the optimal system, we have discussed all possible similarity solutions for inequivalent optimal classes individually. Also, we have found that the similarity solutions exist in two cases for power law shock path and in one case with exponential law shock path. The shock wave decay is due to an increment in the physical parameter such as shock Cowling number or non-idealness parameter or adiabatic exponent or rotational parameter. For smaller value of the ratio of density of solid particles to initial species density of the gas i.e. Ga = 5, the shock strength decreases with an increment in the mass fraction of solid particles in the mixture kp, but the shock strength decreases as kp increases for Ga = 50. Also, the shock strength increases with an increment in Ga at constant kp. It is observed that the shock wave is stronger in the presence of axial magnetic field as compared to azimuthal magnetic field. [ABSTRACT FROM AUTHOR]

Several scientific areas utilize fractional nonlinear partial differential equations (PDEs) to model various phenomena, yet most of these equations lack exact solutions (Ex‐Ss). Consequently, techniques for obtaining approximate solutions (App‐S), which sometimes yield Ex‐Ss, are essential for solving these equations. This study employs a novel technique by combining the residual function and modified fractional power series (FPS) with the Aboodh transform (A‐T) to solve various nonlinear problems within the framework of the Caputo derivative. Studies on absolute error (Abs‐E), relative error (Rel‐E), residual error (Res‐E), and recurrence error (Rec‐E) validate the accuracy and effectiveness of our approach. We apply the limit principle at infinity to determine the coefficients of the series solution terms. In contrast, other methods, such as variational iteration, homotopy perturbation, and Elzaki Adomian decomposition, rely on integration, while the residual power series method (RPSM) employs differentiation, both of which face challenges in fractional scenarios. Moreover, the efficiency of our approach in solving nonlinear problems without depending on Adomian and He polynomials makes it more effective than various approximate series solution techniques. Our method yields results that are very similar to those obtained from the differential transform, the homotopy perturbation, the analytical computational, and Adomian decomposition methods (ADMs). This demonstrates that our technique is a suitable alternative tool for solving nonlinear models. [ABSTRACT FROM AUTHOR]

We argue that the spatial discretization of the strongly nonlinear Lefever–Lejeune partial differential equation defines a nonlinear lattice that is physically relevant in the context of the nonlinear physics of ecosystems, modelling the dynamics of vegetation densities in dry lands. We study the system in the lattice ℤ2$$ {\mathrm{\mathbb{Z}}}^2 $$, which is especially relevant because of its natural dimension for the emergence of pattern formation. Theoretical results identify parametric regimes for the system that distinguish between extinction and potential convergence to nontrivial states. Importantly, we analytically identify conditions for Turing instability, detecting thresholds on the discretization parameter for the manifestation of this mechanism. Numerical simulations reveal the sharpness of the analytical conditions for instability and illustrate the rich potential for pattern formation even in the strongly discrete regime, emphasizing the importance of the interplay between higher dimensionality and discreteness. [ABSTRACT FROM AUTHOR]

Summary: This paper designs an interval observer for switched nonlinear partial differential equation (PDE) systems. Initially, persistent dwell‐time switching rules are used to model switched PDE systems with fast and slow switching phenomena. Next, for unobservable systems caused by uncertainties, the interval observer for the target PDE systems is proposed by utilizing unknown but bounded information of the initial states, boundary conditions, external disturbances, and measurement noises. Subsequently, by utilizing the estimated state's upper and lower bounds, an interval observer‐based control strategy is devised to stabilize the studied systems, and sufficient conditions to ensure the stability of the target systems and the observation error dynamics are provided. Furthermore, the designed interval observer is employed to detect sensor failures in the presence of various uncertainties. Finally, two examples, including the lithium‐ion battery's temperature estimation and fault detection, are utilized to demonstrate the effectiveness of the obtained methods. [ABSTRACT FROM AUTHOR]

The aim of this study is to investigate the entropy production characteristics of two non-miscible fluids in an inclined porous channel with temperature-dependent thermal conductivity and viscosity. The porous region of the channel is divided in two regions. In region-1 and region-2, the Couple stress and Newtonian fluid take place due to constant pressure gradient, respectively, under the influence of a uniform magnetic field. Here, the Darcy–Brinkman model is used for the flow of immiscible fluid through the porous media. In this work, we used a semi-analytical method named as homotopy analysis method (HAM) to solve the coupled nonlinear ordinary differential equations. The goal of the considered problem is to examine the consequences of a variety of thermophysical parameters, including Hartmann number, varying viscosity parameter, varying thermal conductivity parameters, and Grashof number on the characteristics of entropy generation, Bejan number distribution, thermal behavior and flow characteristics of non-miscible couple stress and Newtonian fluid passing through a porous channel. The novel aspect of this study is the formation of entropy and Bejan number as a result of non-miscible Newtonian and couple stress fluids with varying thermal conductivity and viscosity in porous media. In terms of rheological investigation, a semi-analytical simulation for changeable thermal and flow properties in an immiscible Newtonian and couple stress fluid via an inclined porous channel is a brand-new concept, and the behaviors of such flows have not been examined yet. From this study, it is concluded that on raising the variable thermal conductivity, Hartmann number and the permeability of the porous medium, the flow velocity, thermal characteristics and entropy generation number decrease. The authors also come to the significant conclusion that non-miscible Newtonian and couple stress fluids have larger entropy production numbers, flow velocities, and temperature profiles for higher values of Grashof number, variable viscosity parameter, and couple stress parameter. The findings of this work have also been graphically validated through the previously established work. The results of the present analysis can be used in petroleum industry, lubrication theory, etc. [ABSTRACT FROM AUTHOR]

In this paper, the self-propelled movement on gyrotactic swimming microorganisms into this generalized slip flow by nanoliquid over the stretching cylinder with strong magnetic field is discussed. Constant wall temperature was pretended as well as the Nield conditions of boundary. The intuitive non-Newtonian particulate suspension was included into applying Casson fluid by the base liquid and nanoparticles. This formation on the bio-mathematical model gives the boundary value problem by the nonlinear partial differential equations. Primly, modeled numerical system was converted to nondimensional against this help on acceptable scaling variables and the bvp4c technique was used to acquire the mathematical outcomes on the governing system. This graphical description by significant parameters and their physical performance was widely studied. The Prandtl number has the maximum contribution (112.595%) along the selected physical parameters, whereas the Brownian motion has the least (0.00165%) heat transfer rate. Anyhow, Casson fluid was established for much helpful suspension of this method on fabrication and coatings, etc. Therefore, this magnetic field performs like the resistive force of that fluid motion, and higher energy was enlarged into the structure exhibiting strong thermal radiation. [ABSTRACT FROM AUTHOR]

This paper examines the stability analysis and exact solitary wave solutions of the nonlinear partial differential equation known as the Heimburg model. The several types of solitary wave solutions, soliton solutions and Jacobi elliptic doubly periodic function solutions are explored by using the extended Sinh-Gordon equation expansion approach. These investigations exhibit the system's astounding diversity of waveforms, highlighting its potential applications in nerves and biomembranes. By selecting some appropriate values for the parameters, 3D, 2D, and its corresponding contour graph are plotted to represent the physical relevance of some of the solutions. Additionally, the linearized stability of this system is analyzed. The suggested approach is the finest resource for the analytical investigation of any nonlinear issue that occurs in various scientific fields. [ABSTRACT FROM AUTHOR]

This article presents an analytical investigation performed on a generalized geophysical Korteweg–de Vries model with nonlinear power law in ocean science. To start with, achieving diverse solitary wave solutions to the generalized power‐law model involves using wave transformation, which reduces the model to a nonlinear ordinary differential equation. A direct integration approach is adopted to construct solutions in the beginning. This brings the emergence of interesting solutions like non‐topological solitons, trigonometric functions, exponential functions, elliptic functions, and Weierstrass functions in general structures. Besides, in a bid to secure more general exact solutions to the model, one adopts the extended Jacobi elliptic function expansion technique (for some specific cases of n$$ n $$). Thus, various cnoidal, snoidal, and dnoidal wave solutions to the understudied model are attained. The copolar trio explicated in a tabular form reveals that these solutions can be relapsed to various hyperbolic and trigonometric functions under certain criteria. Additionally, diverse graphical exhibitions of the dynamical attributes of the gained results are presented in a bid to have a sound understanding of the physical phenomena of the underlying model. Later, one gives the conserved vectors of the aforementioned equation by employing the standard multiplier approach. [ABSTRACT FROM AUTHOR]

This paper presents a new framework based on nonlinear partial differential equations and statistics. For the nonlinear Phi-4 equation, the probability density function of the hyperbolic secant (HS) distribution has been obtained. Our model's density has various shapes, including left-skewed, symmetric, and right-skewed. Eight distinct estimation approaches have been employed to estimate the parameters of our model. Additionally, the behavior of the HS model parameters was investigated using randomly generated data sets using these estimation techniques. Furthermore, we illustrate the applicability of the HS distribution for modeling real data by applying our results to real data. As a result, it is expected that our proposal will be of significant assistance to the community investigating new distributions based on hyperbolic functions and their applications to real-world data sets. [ABSTRACT FROM AUTHOR]

A signal decomposition is presented that disentangles the deterministic and stochastic components of a multivariate time series. The dynamical component analysis (DyCA) algorithm is based on the assumption that an unknown set of ordinary differential equations (ODEs) describes the dynamics of the deterministic part of the signal. The algorithm is thoroughly derived and accompanied by a link to the GitHub repository containing the algorithm. The method was applied to both simulated and real-world data sets and compared to the results of principal component analysis (PCA), independent component analysis (ICA), and dynamic mode decomposition (DMD). The results demonstrate that DyCA is capable of separating the deterministic and stochastic components of the signal. Furthermore, the algorithm is able to estimate the number of linear and non-linear differential equations and to extract the corresponding amplitudes. The results demonstrate that DyCA is an effective tool for signal decomposition and dimension reduction of multivariate time series. In this regard, DyCA outperforms PCA and ICA and is on par or slightly superior to the DMD algorithm in terms of performance. [ABSTRACT FROM AUTHOR]

The objective of this study is to conduct a numerical examination of the influence of nonlinear chemical reaction and heat source or sink on magnetohydrodynamic (MHD) heat and mass transmission nanofluid flow through a shrinking permeable surface. In addition, the investigation considers thermal radiation and the occurrence of viscous dissipation. Ethylene glycol (EG) is used as the primary fluid medium, whilst the nanoparticles consist of nickel–zinc ferrite. The use of nanofluid flow has garnered significant interest as a result of its potential applications across several sectors. Nanofluids possess a notable benefit in comparison to traditional fluids as a consequence of their enhanced heat transfer capabilities. This advantage may be ascribed to the inclusion of nanoparticles, which augment thermal conductivity and therefore lead to enhanced heat dissipation and efficiency. The mathematical flow model, which is formulated using nonlinear partial differential equations (PDEs), may be transformed into a set of ordinary differential equations (ODEs) by the application of suitable similarity conversions. In order to address the complexities of the nonlinear system, the bvp4c and shooting techniques are used inside the MATLAB program, a widely utilized commercial platform, to effectively solve the associated ODEs by numerical means. This study presents a graphical analysis of the effects of flow parameters on several variables of interest. This research paper presents a numerical examination of the impact of non-linear chemical reactions and heat sources or sinks on magnetohydrodynamic (MHD) heat and mass transfer in nanofluid flow through a shrinking permeable surface. The fluid motion is induced by a stretching sheet within a porous medium. Results indicate a significant influence of stretching sheet geometry on the flow characteristics. Additionally, a comparative analysis with existing literature highlights the benchmark quality of the proposed numerical approach, showcasing its effectiveness and relevance. [ABSTRACT FROM AUTHOR]

This work employs the Extended Direct Algebraic Method (EDAM) to solve quadratic and cubic nonlinear Klein–Gordon Equations (KGEs), which are standard models in particle and quantum physics that describe the dynamics of scaler particles with spin zero in the framework of Einstein's theory of relativity. By applying variables-based wave transformations, the targeted KGEs are converted into Nonlinear Ordinary Differential Equations (NODEs). The resultant NODEs are subsequently reduced to a set of nonlinear algebraic equations through the assumption of series-based solutions for them. New families of soliton solutions are obtained in the form of hyperbolic, trigonometric, exponential and rational functions when these systems are solved using Maple. A few soliton solutions are considered for certain values of the given parameters with the help of contour and 3D plots, which indicate that the solitons exist in the form of dark kink, hump kink, lump-like kink, bright kink and cuspon kink solitons. These soliton solutions are relevant to actual physics, for instance, in the context of particle physics and theories of quantum fields. These solutions are useful also for the enhancement of our understanding of the basic particle interactions and wave dynamics at all levels of physics, including but not limited to cosmology, compact matter physics and nonlinear optics. [ABSTRACT FROM AUTHOR]

In this paper, we develop a hybrid approach to solve the viscous Burgers' equation by combining classical boundary layer theory with modern Physics-Informed Neural Networks (PINNs). The boundary layer theory provides an approximate analytical solution to the equation, particularly in regimes where viscosity dominates. PINNs, on the other hand, offer a data-driven framework that can address complex boundary and initial conditions more flexibly. We demonstrate that PINNs capture the key dynamics of the Burgers' equation, such as shock wave formation and the smoothing effects of viscosity, and show how the combination of these methods provides a powerful tool for solving nonlinear partial differential equations. [ABSTRACT FROM AUTHOR]

The purpose of this article concerns a type of nonlinear boundary value problem of fractional differential equations with the Caputo derivative. Using the fixed point index theory of condensing mapping in cones, the existence results of positive solutions for such a problem are obtained. [ABSTRACT FROM AUTHOR]

The Wazwaz–Benjamin–Bona–Mahony (WBBM) equation is a well-known regularized long-wave model that examines the propagation kinematics of water waves. The current work employs an effective approach, called the Riccati Modified Extended Simple Equation Method (RMESEM), to effectively and precisely derive the propagating soliton solutions to the (3+1)-dimensional WBBM equation. By using this upgraded approach, we are able to find a greater diversity of families of propagating soliton solutions for the WBBM model in the form of exponential, rational, hyperbolic, periodic, and rational hyperbolic functions. To further graphically represent the propagating behavior of acquired solitons, we additionally provide 3D, 2D, and contour graphics which clearly demonstrate the presence of kink solitons, including solitary kink, anti-kink, twinning kink, bright kink, bifurcated kink, lump-like kink, and other multiple kinks in the realm of WBBM. Furthermore, by producing new and precise propagating soliton solutions, our RMESEM demonstrates its significance in revealing important details about the model behavior and provides indications regarding possible applications in the field of water waves. [ABSTRACT FROM AUTHOR]

This article is devoted to the mathematical analysis of a heat and mass transfer model for the pressure-induced flow of a viscous fluid through a plane channel subject to Navier's slip conditions on the channel walls. The important feature of our work is that the used model takes into account the effects of variable viscosity, thermal conductivity, and slip length, under the assumption that these quantities depend on temperature. Therefore, we arrive at a boundary value problem for strongly nonlinear ordinary differential equations. The existence and uniqueness of a solution to this problem is analyzed. Namely, using the Galerkin scheme, the generalized Borsuk theorem, and the compactness method, we proved the existence theorem for both weak and strong solutions in Sobolev spaces and derive some of their properties. Under extra conditions on the model data, the uniqueness of a solution is established. Moreover, we considered our model subject to some explicit formulae for temperature dependence of viscosity, which are applied in practice, and constructed corresponding exact solutions. Using these solutions, we successfully performed an extra verification of the algorithm for finding solutions that was applied by us to prove the existence theorem. [ABSTRACT FROM AUTHOR]

The collocation method is one of the most powerful and effective techniques to solve nonlinear differential equations. This method reduces the original problem to a system of algebraic equations. Awareness of the nature of the problem and the analytical behavior of the solution has an important influence on the choice of type and number of basis functions required by the collocation method for solving the different classes of differential equations. In general, the polynomial basis functions such as the Legendre polynomials are not always an appropriate choice to approximate the functions that can be expanded based on monomial basis functions with real powers. In such cases, a combination of polynomial basis functions with some maps to replace the natural powers with arbitrary real powers can be a suitable alternative. This paper presents a collocation method based on the shifted fractional-order Legendre functions to solve the linear and nonlinear Hamilton–Jacobi–Bellman PDEs in stochastic optimal control problems. The convergence analysis of the proposed method has also been investigated. Moreover, the method is used to solve some practical problems with different solution structures such as resource extraction and Merton's portfolio selection models. Numerical results for four different examples indicated that the collocation method based on the fractional-order Legendre functions are provided. Due to the non differentiability of the solution of HJB PDEs at x = 0 for resource extraction and Merton's portfolio selection models, the Legendre collocation method cannot produce solutions with the desired accuracy. In fact, it is observed that the collocation method based on the fractional-order Legendre functions can be more efficient than the standard Legendre collocation method. Also, the sample paths of the control and state processes are obtained together with the estimate given by the Monte Carlo simulation. [ABSTRACT FROM AUTHOR]

This work investigates the Kraenkel-Manna-Merle (KMM) system, which models the nonlinear propagation of short waves in saturated ferromagnetic materials subjected to an external magnetic field, despite the absence of electrical conductivity. The study aims to explore and derive new solitary wave solutions for this system using two distinct methodological approaches. In the first approach, the KMM system is transformed into a system of nonlinear ordinary differential equations (ODEs) via Lie group transformation. The resulting ODEs are then solved analytically using a similarity invariant approach, leading to the discovery of various types of solitary wave solutions, including bright, dark, and exponential solitons. The second approach involves applying wave and Galilean transformations to reduce the KMM system to a system of two ODEs, both with and without damping effects. This reduced system is further analyzed to investigate its bifurcation behavior, sensitivity to initial conditions, and chaotic dynamics. The analysis reveals the presence of strange multi-scroll chaotic dynamics in the presence of damping and off-boosting dynamics without damping. In addition to these approaches, the study also applies the planar dynamical theory to obtain further new soliton solutions of the KMM system. These solitons include bright, kink, dark, and periodic solutions, each of which has been visualized through 3D and 2D graphs. The results of this research provide new insights into the dynamics of the KMM system, with potential applications in magnetic data storage, magnonic devices, material science, and spintronics. [ABSTRACT FROM AUTHOR]

This study presents, for the first time, a novel mathematical model that can address the nonlinear buckling, postbuckling, and snap-through of bidirectional functionally graded porous simply supported plates rested on elastic foundation and subjected to uniaxial/biaxial compressive loads. The parabolic shear deformation plate theory and the von Kármán strain nonlinearity are employed to derive governing equilibrium equations relative to neutral surface plane. The governing equations that consist of four nonlinear-coupled variable-coefficients partial differential equations are discretized by using the differential/integral quadrature method. The solution methodology depends on whether the discretized nonlinear algebraic system is homogeneous (with zero force vector) or nonhomogeneous. A homogeneous system can be formulated and solved as a nonlinear eigenvalue problem. Pseudo-arc-length continuation is implemented with a proposed iterative method to predict the load-deflection paths whether the system is homogeneous or not. Theoretical analysis and numerical results indicate that the type of porosity and position of in-plane loading have significant effects on the pitchfork-bifurcation or snap-through instability response of the bidirectional functionally graded porous plate. Parametric studies are presented to illustrate the impact of gradation indices, geometrical properties, porosity, and foundation constants on the postbuckling responses of bidirectional functionally graded porous plates. The proposed model may be used in designing nuclear, marine, aerospace, and civil structures with bi-directional functionally graded material constituents. [ABSTRACT FROM AUTHOR]

Purpose: The main purpose of this study is to analyze the heat transfer phenomena in a dynamically bulging enclosure filled with Cu-water nanofluid. This study examines the convective heat transfer process induced by a bulging area considered a heat source, with the enclosure's side walls having a low temperature and top and bottom walls being treated as adiabatic. Various factors, such as the Rayleigh number (Ra), nanoparticle volume fraction, Darcy effects, Hartmann number (Ha) and effects of magnetic inclination, are analyzed for their impact on the flow behavior and temperature distribution. Design/methodology/approach: The finite element method (FEM) is employed for simulating variations in flow and temperature after validating the results. Solving the non-linear partial differential equations while incorporating the modified Darcy number (10−3 ≤ Da ≤ 10−1), Ra (103 ≤ Ra ≤ 105) and Ha (0 ≤ Ha ≤ 100) as the dimensionless operational parameters. Findings: This study demonstrates that in enclosures with dynamically positioned bulges filled with Cu-water nanofluid, heat transfer is significantly influenced by the bulge location and nanoparticle volume fraction, which alter flow and heat patterns. The varying impact of magnetic fields on heat transfer depends on the Rayleigh and Has. Practical implications: The geometry configurations employed in this research have broad applications in various engineering disciplines, including heat exchangers, energy storage, biomedical systems and food processing. Originality/value: This research provides insights into how different shapes of the heated bulging area impact the hydromagnetic convection of Cu-water nanofluid flow in a dynamically bulging-shaped porous system, encompassing curved surfaces and various multi-physical conditions. [ABSTRACT FROM AUTHOR]

Purpose: In this study, we investigate the effects of an extended ternary hybrid Tiwari and Das nanofluid model on ethylene glycol flow, with a focus on heat transfer. Using the Cross non-Newtonian fluid model, we explore the heat transfer characteristics of this unique fluid in various applications such as pharmaceutical solvents, vaccine preservatives, and medical imaging techniques. Design/methodology/approach: Our investigation reveals that the flow of this ternary hybrid nanofluid follows a laminar Cross model flow pattern, influenced by heat radiation and occurring around a stretched cylinder in a porous medium. We apply a non-similarity transformation to the nonlinear partial differential equations, converting them into non-dimensional PDEs. These equations are subsequently solved as ordinary differential equations (ODEs) using MATLAB's bvp4c tools. In addition, the magnetic number in this study spans from 0 to 5, volume fraction of nanoparticles varies from 5% to 10%, and Prandtl number for EG as 204. This approach allows us to examine the impact of temperature on heat transfer and distribution within the fluid. Findings: Graphical depictions illustrate the effects of parameters such as the Weissenberg number, porous parameter, Schmidt number, thermal conductivity parameter, Soret number, magnetic parameter, Eckert number, Lewis number, and Peclet number on velocity, temperature, concentration, and microorganism profiles. Our results highlight the significant influence of thermal radiation and ohmic heating on heat transmission, particularly in relation to magnetic and Darcy parameters. A higher Lewis number corresponds to faster heat diffusion compared to mass diffusion, while increases in the Soret number are associated with higher concentration profiles. Additionally, rapid temperature dissipation inhibits microbial development, reducing the microbial profile. Originality/value: The numerical analysis of skin friction coefficients and Nusselt numbers in tabular form further validates our approach. Overall, our findings demonstrate the effectiveness of our numerical technique in providing a comprehensive understanding of flow and heat transfer processes in ternary hybrid nanofluids, offering valuable insights for various practical applications. [ABSTRACT FROM AUTHOR]

In this paper, the dynamic behavior of one degree-of-freedom oscillator subject to stick–slip and wear phenomena at the contact interface with a rigid substrate is investigated. The motion of the oscillator, induced by a harmonic excitation, depends on the tangential contact forces, exchanged with the rigid soil, which are modeled through piecewise nonlinear constitutive laws, accounting for stick–slip phenomena due to friction as well as wear due to abrasion, already developed by the authors in a previous work. The nonlinear ordinary differential equations governing the problem are derived, whose solution is numerically obtained via a typical Runge–Kutta-based algorithm. The main target of this study is to analyze and discuss the strong nonlinear behavior, descending from the presence of stick–slip and wear phenomena, thus investigating the effect of the different interface modeling. In this framework, the analysis is carried out considering the whole evolution of non-smooth contact laws, starting from the virgin interface. [ABSTRACT FROM AUTHOR]

In this paper we consider two classes of nonlinear partial differential equations with the fractional Laplacian, namely $$\begin{align*} \left(-\Delta\right)^{\frac{\alpha}{2}}\left(u^{m} \right) &=u\left|u\right|^{q-1}+w\left(x\right), \quad x\in \mathbb{R}^{N},\\ &\qquad 1\leq m (− Δ) α 2 (u m) = u | u | q − 1 + w (x) , x ∈ R N , 1 ≤ m < q , 0 < α ≤ 2 , N > α and $$\begin{align*} &\frac{\partial^{k}u}{\partial t^{k}}+\left(-\Delta\right)^{\frac{\alpha}{2}}u=u^{q}, \quad \left(x,t\right)\in \mathbb{R}^{N}\times\left(0,+\infty\right),\\ &\qquad k\geq 1, \ 0 ∂ k u ∂ t k + (− Δ) α 2 u = u q , (x , t) ∈ R N × (0 , + ∞) , k ≥ 1 , 0 < α ≤ 2 , N > α. Solutions defined for all $ x\in \mathbb {R}^{N} $ x ∈ R N of the first equation are referred to as entire solutions, while solutions defined for all $ \left (x,t\right)\in \mathbb {R}^{N}\times \left [0,+\infty \right) $ (x , t) ∈ R N × [ 0 , + ∞) of the second equation are referred to as global solutions. Several existence and nonexistence theorems are established over different ranges of q, and thus the respective relations between the existence, nonexistence of solutions for these equations and the index q in the nonlinear terms are obtained. It is illustrated that our results are sharp in cases of m = 1 and k = 1 respectively. In addition, we prove the positivity, symmetry and odevity of solutions we constructed for the first equation with m = 1 associated with the inhomogeneous term w. [ABSTRACT FROM AUTHOR]

The purpose of this article is to study the effect of electromagnetic (EM) stimulation on the pipeline, which has an electrical and thermal effect in addition to the chemical reaction on the crude oil and makes a sinusoidal wave on the wall. Modeling the crude oil as Carreau fluid is done. EM stimulation is an effective and safe technology that can be used to improve fluid movement in a variety of industrial applications. The flow analysis by applying EM may avoid blocking the crude oil pipeline which leads to a loss of production and capital investment. The basic partial differential equations of momentum, temperature and concentration are reduced to a system of nonlinear partial differential equations, which is solved numerically by using the Rung–Kutta–Merson method with Newton iteration in a shooting and matching technique under the assumption of long wavelength and the effect of physical implanted parameters is represented through charts for velocity, temperature, and concentration and numerical application. [ABSTRACT FROM AUTHOR]

The elastic–plastic infinitesimal deformation of a solid is considered within the framework of the incremental flow theory using the constitutive relation in the general rate form. The appropriate initial boundary value problem is formulated for the displacement in the form of the evolutionary-variational problem (EVP), i.e., as the abstract Cauchy problem in the Hilbert space which coincides with a weak form of the equilibrium equation, known as the principle of possible displacements in mechanics. The general existence and uniqueness theorem for the EVP is discussed. The main sufficient condition has a simple algebraic form and does not coincide with the classical Drucker and similar thermodynamical postulates; therefore, it must be independently verified. Its independence is illustrated for the non-associated plastic model of linear isotropic-kinematic hardening with dilatation and internal friction. The classical and endochronic models are analyzed too. The initial EVP is reduced by a spatial finite element approximation to the Cauchy problem for an implicit system of essentially nonlinear ordinary differential equations which can be stiff. Therefore, for the numerical solution the implicit Euler scheme is proposed. All theoretical results are illustrated by means of original numerical experiments. [ABSTRACT FROM AUTHOR]

A nanofluid refers to the dispersion of nanoparticles in a regular fluid and has a unique application in various sectors, including medicine, engineering, and technology. When multiple nanoparticles are suspended in a regular fluid, it creates a hybrid nanofluid. In this study, we aim to investigate homogenous–heterogenous reactions in Bödewadt hybrid nanofluid flow over a permeable rotating disk with radiation. The base fluid chosen for this study is water (H2O), while the nanoparticles iron oxide (Fe3O4) and cobalt ferrite (CoFe2O4) are utilized to create the hybrid nanofluid. An appropriate method of similarity transformation is executed along a set of partial differential equations (PDEs) that were reduced to a system of nonlinear ordinary differential equations (ODEs). Numerical outcomes were then obtained via bvp4c in MATLAB software, with the influence of various parameters such as nanoparticle volume fraction, homogenous/heterogenous reaction strength parameters, suction, shrinking/stretching parameters, and radiation parameter. Additionally, asymptotic analysis was conducted to show that the concentration boundary layer on the disk can be performed subject to a large number of suctions. The present findings reveal that a rise in the volume fraction of nanoparticles results in a reduction in radial velocity profiles, temperature profiles, and tangential fields. As thermal radiation levels rise, a notable reduction in the local Nusselt number is evident. Moreover, there is an observed linear escalation in wall surface concentration when the heterogeneous strength parameter attains higher values. The presented results demonstrate that all flow fields are significantly affected by the participating parameters. [ABSTRACT FROM AUTHOR]

Chlorinated organic compounds are Persistent Organic Pollutants (POPs) with high environmental continuity. These chemicals possess the ability to permeate into the environment across both unsaturated and saturated zones. This study examines the potential impact of perchloroethylene (PCE) releases in aquifer systems consisting of layers with varying permeability. The numerical simulations utilized the CactusHydro numerical code that employs a high-resolution shock-capturing flux conservative method to solve the non-linear partial differential equations of a three-phase immiscible fluid flow and study the migration of PCE into variably saturated zones to predict the spatial and temporal distribution of free products across multilayered aquifers. Two scenarios were simulated to predict how a first low-permeable horizon would affect the downward migration of PCE in its pure phase. The numerical simulations show that the multilayered aquifer system is vulnerable to an in-depth study of the PCE migration paths in the pure phase. However, very long times (about a year) are required for the DNAPL to be able to cross the semi-permeable horizons. The results in both scenarios show the quantity of mass (in kg and percentage concerning the initial one) that is left along the multilayered aquifer during the migration and after several years. [ABSTRACT FROM AUTHOR]

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In this paper, we prove the existence of multiple solutions in the Bessel Potential space for a new class of nonlinear fractional Schr¨odinger-Poisson systems involving the distributional Riesz fractional derivative. To reach our goal, we use the symmetric mountain pass theorem under some suitable assumptions on nonlinearity f(x, u) and potential V (x). [ABSTRACT FROM AUTHOR]

In this paper, methodological approaches for distribution functions of nonlinear stochastic differential equations (SDEs) with uncertain parameters are elaborated. The stationary and nonstationary Fokker–Planck–Kolmogorov (FPK) equations associated with the considered random equations are investigated. A semi-analytical approach based on the extended exponential closure handling the parameters uncertainty effects is presented. A new numerical method coupling the polynomial chaos with the differential quadrature method (DQM) is elaborated for the stationary case. A compact matrix formulation is established allowing considering a large number of generalized polynomial chaos. For nonstationary random FPK equations, the polynomial chaos is coupled with the quadrature discretization. A symmetric discretization of the main Fokker–Planck operator using a tensor product of the Hilbert state space and the Hilbert probabilistic space has resulted. The obtained eigenvalues and random eigenfunctions are used for a spectral decomposition. The time solution is obtained by the spectral expansion as well as by the random Euler stochastic method. The accuracy, effectiveness and advantages of the developed procedure in analyzing the stationary and nonstationary probabilistic solutions of nonlinear stochastic dynamical systems with uncertain parameters excited by a Gaussian white noise are demonstrated. [ABSTRACT FROM AUTHOR]

This study focuses on investigating the effects of viscous dissipation, magnetohydrodynmaics (MHD) and mixed convection on Casson nanofluid thin film flow over a vertical stretching surface, considering its unsteady, two‐dimensional nature. The problem is formulated by assuming incompressible flow and employing boundary layer approximations, leading to a system of partial differential equations (PDEs). These PDEs are then transformed into nonlinear ordinary differential equations (ODEs) using similarity transformations. Numerical techniques, facilitated by computational software such as Mathematica, are employed to solve these nonlinear ODEs. The investigation probes into the mathematical analysis of various physical parameters, presenting their effects through graphical representations. The impact of parameters such as concentration Grashof number (Gc), thermal Grashof number (Gr), magnetic parameter (M), unsteadiness parameter (S), dimensionless film thickness (β), and Casson fluid parameter (γ) are examined. The velocity profile f′(η)$f^{\prime}( \eta )$ exhibits an increase with rising values of Gc and Gr, while it decreases with increasing values of M, S, β, and γ. The results are further elucidated through tabular representations, providing insights into the variations of skin‐friction coefficient, Sherwood number, and Nusselt number with respect to the aforementioned parameters. [ABSTRACT FROM AUTHOR]

A computational mechanism has been formulated to scientifically analyze the phenomenon of combined convection in the occurrence of magnetohydrodynamic flow behavior of Casson nanofluid. This study emphasizes the trajectory of the nanofluid along a stretching sheet with nonlinear permeability, while considering additional physical effects such as reversible chemical reaction, thermal radiation, energy source/sink, suction and viscous dissipation. In this study, we have additionally integrated the nanofluid paradigm proposed by Buongiorno, which encompasses the repercussion of thermophoresis along with Brownian motion. The utilization of appropriate similarity renovations serves the purpose of recasting the dominant multivariate differential equations to a collection of nonlinear differential equations in single variable. The Runge–Kutta shooting mechanism is implemented for the intention of numerically determining the unknown boundary conditions. The solution to the dominant equations was obtained by employing the fourth-order Runge–Kutta technique and to ensure the dependability of the outcomes, the bvp4c method was employed. The graphical and tabulated observations are presented herein to facilitate a comprehensive analysis of the underlying physical characteristics inherent in the problem at hand. The utilization of the Casson parameter has been found to be advantageous in the reduction of shear stress rate, along with the enhancement of mass and energy transfer rates. Additionally, the application of suction has been observed to be beneficial in the enhancement of Sherwood number and energy transfer rate. Within the scope of the context of the esterification process, the purpose of this proposal is to evaluate the impact that numerous arithmetical values exert on the temperature field, the velocity profile and the volumetric concentration. An in-depth graphical analysis that takes into account the relevant physical consequences is used to evaluate the most important factors that relate to the physical quantities that describe the contours of temperature, velocity and concentration. In addition to being accompanied by their respective explanations, the tabular portrayal of the local Sherwood number, shear stress rate along with local Nusselt number all feature in this paper. There is a clear distinction that can be made between irreversible and reversible flows in the evaluation of the local Sherwood number, rate of shear stress and local Nusselt number. This differentiation is brought about by taking into analysis thermophoresis parameter, Brownian motion parameter and suction parameter. The results that were produced from the theoretical simulations have significant repercussions for a variety of fields that are related to energy engineering. The findings derived from the analysis indicate a negative correlation between the Brownian motion parameter and both irreversible and reversible flow in terms of rate of energy transfer and shear stress rate. It is imperative to highlight that reversible flow holds greater significance in comparison to irreversible flow. [ABSTRACT FROM AUTHOR]

This paper examines the entropy generation and magneto-Soret effects on the convective-radiative heat transport of a chemically reactive fluid past an oscillatory surface in a permeable media with an internal heat source and suction at the porous wall. The perturbation scheme is adopted to convert the basic nonlinear partial differential equations into a set of linear ordinary differential equations that are solved analytically. It is found that the viscous dissipative force and the thermal radiation parameter both reduce the concentration distribution within the fluid. Physically, higher thermal radiation can increase the temperature within the fluid, leading to a decrease in concentration distribution. Conversely, an increase in the Soret parameter, which accounts for the effect of temperature gradients on mass diffusion, results in an opposite trend, enhancing the concentration distribution. The Soret effect, or thermal diffusion, causes mass to move from regions of high temperature to low temperature. Additionally, the total entropy of the fluid is examined across various values of the Reynolds number, Brinkman number, temperature difference parameter, and magnetic parameter. As the magnetic parameter increases, the entropy within the system rises, indicating enhanced irreversibilities and energy dissipation due to the presence of a magnetic field. [ABSTRACT FROM AUTHOR]

ABSTRACT Unsaturated zones are important for geotechnical design, geochemical reactions, and microbial reactions. The numerical analysis of unsaturated seepage is complex because it involves highly nonlinear partial differential equations. The permeability can vary by orders of magnitude over short vertical distances. This article defines and uses H‐convergence tests to quantify numerical errors made by uniform meshes with element size (ES) for 1D steady‐state conditions. The quantitative H‐convergence should not be confused with a qualitative mesh sensitivity study. The difference between numerical and mathematical convergences is stated. A detailed affordable method for an H‐convergence test is presented. The true but unknown solution is defined as the asymptote of the numerical solutions for all solution components when ES decreases to zero. The numerical errors versus ES are then assessed with respect to the true solution, and using a log–log plot, which indicates whether a code is correct or incorrect. If a code is correct, its results follow the rules of mathematical convergence in a mathematical convergence domain (MCD) which is smaller than the numerical convergence domain (NCD). If a code is incorrect, it has an NCD but no MCD. Incorrect algorithms of incorrect codes need to be modified and repaired. Existing codes are shown to converge numerically within large NCDs but generate large errors, up to 500%, in the NCDs, a dangerous situation for designers. [ABSTRACT FROM AUTHOR]

This paper presents a modification of the Differential Transform Method (ModDTM) formulated so that the search direction of the solution is along a spatial axis (x). It is shown that the method is applicable to a wide spectrum of (1 + 1) partial differential equations, integro partial differential equations, and integral equations. The solutions obtained are in the form of a Taylor series, the coefficients of which are determined by recursively operating a differential transform of the equation. These solutions are either exact and in closed form, or are good approximations. To illustrate the application of the method, as well as to show its versatility, examples are chosen from all the categories mentioned earlier. These include equations such as the nonlinear Fisher, combined KdV‐mKdV, Hunter–Saxton (H‐S), Fornberg–Whitham (FW), coupled systems such as the Whitham–Broer–Kaup (WBK) and the sine‐Gordon (s‐G) equations, integral equations, Volterra partial integro‐differential equations (PIDE), and linear and nonlinear versions of the (complex) Schröedinger equation. The Tables and 3D‐plots show the good rate of convergence of the obtained solutions with the exact ones. It is thus found that the new scheme is successful in producing satisfactory results as any other method in which the conventional approach of searching for solutions along the temporal axis is followed. Further, the procedures involved are simple, and hence may be operated with great ease. [ABSTRACT FROM AUTHOR]

This study use numerical method to investigate the transportation of a bioconvective magnetized tangent hyperbolic nanofluid across an exponentially porous stretched sheet. The flow model takes into account the important contributions of Joule heating, thermal radiation, activation energy, and nonuniform heat source/sink. Moreover, slip boundary conditions with gyrotactic microorganisms are taken into flow analysis. The flow model is converted into a system of nonlinear ordinary differential equations (ODEs) using suitable similarity variables. The bvp4c approach in MATLAB is used to get numerical solutions for this system. The study evaluates the influence of different parameters on the velocity, temperature, concentration, and microorganism profile via graphical representations. The results indicate that increasing the mixed convection parameters accelerates the velocity profile, whereas raising the magnetic parameter slows it down. The temperature profile increases with higher values of radiation parameter, however, it decreases as the Prandtl number and thermal slip parameter increase. Furthermore, the microorganism profile decreases as the bioconvection Lewis number and microorganism slip parameter increase. [ABSTRACT FROM AUTHOR]

Solving differential equations is an ancient and very important research topic in theory and practice. The exact analytical solution to differential equations can describe various physical phenomena such as vibration and propagation wave. In this paper, the bilinear neural network method (BNNM), which uses neural network to unify all kinds of classical test function methods, is employed to obtain some new exact analytical solutions of the (3 + 1)-dimensional Hirota bilinear (HB) equation. Based on the Hirota form of the HB equation, we constructed four kinds of new solutions which contain the breather solution, rogue wave solution, breather lump-type soliton solution and the interaction solution between the periodic waves and two-kink wave by introducing a series of test functions in both single-layer and multi-layer neurons such as [4–2–2] and [4–2–3] neural network models. In addition, we compared them to those that had already been published. It is clear that our results are not consistent with those found in these publications. Their corresponding dynamic features are vividly demonstrated in some 3D, contour, x-curves and y-curves plots. The results obtained demonstrate the potential of the proposed methods to solve other nonlinear partial differential equations in fields. [ABSTRACT FROM AUTHOR]

AbstractInvestigation of nanofluid flows under various conditions is becoming fascinating research due to its numerous applications in many science and engineering fields such as solar energy, geothermal energy, water purification, glass blowing, enhanced oil recovery, petroleum production and food processing. For the present requirements in thermal energy applications, this mathematical model is considered to explore the diffusion and heat source effects on the flow of conductive Casson and Maxwell nanofluids in a porous medium on an elongating sheet with a chemical reaction. Further, the effects of viscous dissipation and thermal radiation are also examined. Moreover, the entropy generation is analyzed since this approach is employed in solar energy exchangers. The governing equations and the corresponding boundary conditions are constructed. This system of partial differential equations (PDEs) is changed into a system of nonlinear ordinary differential equations (ODEs) with the help of a suitable transformation and then solved numerically by the bvp5c MATLAB package. The influences of the physical parameters on the flow phenomena are discussed and presented through figures and tables. This analysis explains that the Casson fluid flow has a higher velocity field than the Maxwell fluid flow. The Nusselt number becomes higher as the Dufour numbers increase. It has also been noticed that increasing the Brinkmann number improves the entropy and, an increase in radiation enriches the temperature profile. The concentration of the fluid is decreased for the increasing values of the higher-order chemical reaction parameter. Furthermore, comparisons of the current results were conducted for limiting examples of the problem and found to be in good agreement with existing results in the literature. [ABSTRACT FROM AUTHOR]

Nonlinear fractional-order partial differential equations play an important role in science and engineering by illustrating a variety of nonlinear processes. The nonlinear space-time fractional combined Korteweg-de Vries and modified Korteweg-de Vries equation is a very significant fractional partial differential equation and is used to simulate shallow water surface wave phenomena, pulse waves in large arteries, ion acoustic waves in plasmas, and atmospheric dust-acoustic solitary waves. The improved Bernoulli sub-equation function method and the new generalized (G′/G)-expansion method are two noteworthy approaches that have been used to analyze and extract solutions to the above-mentioned equation that include various types of traveling waves as well as soliton solutions via beta-derivative. Through the utilization of a wave transformation, the fractional-order equation is transformed into a nonlinear ordinary differential equation. The exponential function, trigonometric function, rational function, and hyperbolic trigonometric function solutions with arbitrary constants have been used to articulate the obtained solutions. By utilizing the aforementioned approaches, several standard waveforms have been recognized including multiple periodic types, kink shapes, bell-shaped, single solitons, and other types of solitons. Mathematica software has been used to illustrate the wave profiles through 3D and contour plots, providing a clearer physical sketch based on diverse values of free parameters. The suggested methods have demonstrated their reliability in establishing more generalized wave solutions and exhibit computational efficiency, making them suitable for soliton solutions. [ABSTRACT FROM AUTHOR]

Despite the progress in medical data collection the actual burden of SARS‐CoV‐2 remains unknown due to under‐ascertainment of cases. This was apparent in the acute phase of the pandemic and the use of reported deaths has been pointed out as a more reliable source of information, likely less prone to under‐reporting. Since daily deaths occur from past infections weighted by their probability of death, one may infer the total number of infections accounting for their age distribution, using the data on reported deaths. We adopt this framework and assume that the dynamics generating the total number of infections can be described by a continuous time transmission model expressed through a system of nonlinear ordinary differential equations where the transmission rate is modeled as a diffusion process allowing to reveal both the effect of control strategies and the changes in individuals behavior. We develop this flexible Bayesian tool in Stan and study 3 pairs of European countries, estimating the time‐varying reproduction number (Rt$$ {R}_t $$) as well as the true cumulative number of infected individuals. As we estimate the true number of infections we offer a more accurate estimate of Rt$$ {R}_t $$. We also provide an estimate of the daily reporting ratio and discuss the effects of changes in mobility and testing on the inferred quantities. [ABSTRACT FROM AUTHOR]

This article aims to explore the existence and stability of solutions to differential equations involving a ψ -Hilfer fractional derivative in the Caputo sense, which, compared to classical ψ -Hilfer fractional derivatives (in the Riemann–Liouville sense), provide a clear physical interpretation when dealing with initial conditions. We discovered that the ψ -Hilfer fractional derivative in the Caputo sense can be represented as the inverse operation of the ψ -Riemann–Liouville fractional integral, and used this property to prove the existence of solutions for linear differential equations with a ψ -Hilfer fractional derivative in the Caputo sense. Additionally, we applied Mönch's fixed-point theorem and knowledge of non-compactness measures to demonstrate the existence of solutions for nonlinear differential equations with a ψ -Hilfer fractional derivative in the Caputo sense, and further discussed the Ulam–Hyers–Rassias stability and semi-Ulam–Hyers–Rassias stability of these solutions. Finally, we illustrated our results through case studies. [ABSTRACT FROM AUTHOR]

Globally positive unbounded solutions, with zero derivative at infinity, are here considered for ordinary differential equations involving the generalized Euclidean mean curvature operator. When p ≥ 2 , the results highlight an analogy with an auxiliary equation with the p-Laplacian operator. The results are obtained using some comparison criteria for the principal solutions of a class of associated half-linear equations. [ABSTRACT FROM AUTHOR]

This paper explores the oceanic wave characteristics exhibited by a modified integrable generalized (2+1)-dimensional nonlinear Schrödinger system of equations through variable coefficients. Two newly modified methods, specifically the improved nonlinear Ricatti equation method and the improved sub-equation method, have been proposed to investigate the aforementioned nonlinear system. Through the utilization of these methods, we successfully obtain traveling and solitary waves solutions for this nonlinear system. We emphasize several constraint conditions that serve to guarantee the existence of these solutions. More comprehensive information about the physical dynamical representation of some of the solutions presented is illustrated through graphical depictions. The Mathematica software package is employed to produce both three-dimensional and their corresponding contour plots, thereby improving the visualization and comprehension of the solutions. This paper illustrates that the two proposed approaches provide straightforward and efficient means of acquiring various types of soliton, rational, trigonometric, hyperbolic and exponential solutions. Moreover, they present a more potent mathematical tool for addressing a variety of other nonlinear partial differential equations that hold significance in the field of applied science and engineering. [ABSTRACT FROM AUTHOR]