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This paper presents a technique to find the closed‐form solution of the boundary‐value problem representing free convection flow past a vertical cone. Partial differential equations that govern the flow are converted into nonlinear ordinary differential equations with the aid of similarity transformations. First, a numerical solution is obtained with the help of the bvp4c package of MATLAB, and then utilizing the numerical solution, the closed‐form solution is obtained. The accuracy of the closed‐form solution is analyzed by the residual technique. Graphs of the residuals of the governing equations indicate that our approximate closed‐form solution closely matches the exact solution. [ABSTRACT FROM AUTHOR]

The present study aims to quantify the flow field, flow velocity, and heat transfer features over a horizontal flat plate under the influence of an applied magnetic field, with a particular emphasis on low Prandtl number fluids. Nonlinear partial differential expressions can be incorporated into the ordinary differential framework with the use of appropriate transformations. This research utilizes the variational iteration method (VIM) to approximate solutions for the system of nonlinear differential equations that define the problem. The objective is to demonstrate superior flexibility and broader application of the VIM in addressing heat transfer issues, compared to alternative approaches. The results obtained from the VIM are compared with numerical solutions, revealing a significant level of accuracy in the approximation. The numerical findings strongly suggest that the VIM is effective in providing precise numerical solutions for nonlinear differential equations. The analysis includes an examination of the flow field, velocity, and temperature distribution across various parameters. The study found that improving temperature patterns, velocity distribution, and flow dynamics were all positively impacted by increasing the Prandtl numbers. As a result, this leads to the thickness of the boundary layer to decrease and improves heat transfer at the moving surface. Thus, the convection process becomes more efficient. When the strength of the magnetic field is increased, the velocity of the fluid decreases. This observation aligns with expectations since the magnetic field hampers the natural flow of convection. Notably, the convection process can be precisely controlled by carefully applying magnetic force. [ABSTRACT FROM AUTHOR]

The Jeffery–Hamel flow phenomenon appears in a variety of real‐world applications involving the flow of two nonparallel plates. BY using a similarity transformation derived from the equation of continuity, partial differential equations determining flow characteristics are translated into nonlinear ordinary differential equations. The problem involves the flow of a specific type of fluid, namely, an incompressible and electrically conducting fluid, between two nonparallel plates. The flow is assumed to be steady, two‐dimensional, and subject to certain boundary conditions. Specifically, the plates are impermeable, and the fluid adheres to a no‐slip condition, resulting in zero fluid velocity at the plates' surfaces. Moreover, the problem incorporates the effects of magnetic fields and pressure fluctuations, making it highly applicable to scenarios, such as blood flow through arteries in the human body, which can be modeled as a special case of the magnetohydrodynamic (MHD) Jeffery–Hamel problem referred to as the (MHD) blood pressure equation. This work compares two numerical approaches for solving the MHDs Jeffery–Hamel problem: B‐spline and Bernstein polynomial collocation. The given approaches are used to discretize and transform the equation into a system of algebraic equations. Matrix algebra techniques are then used to solve the resultant system. A complete error analysis and convergence rates for different grid sizes are derived for both methods and are used to compare the accuracy and efficiency of the two approaches. Both approaches produce correct solutions, according to the numerical findings, although the Bernstein polynomial collocation method is more efficient and accurate than the B‐spline collocation. [ABSTRACT FROM AUTHOR]

The investigation is about the effect of heat and mass transportation over a dragging out cylinder for Cross fluid flow in addition to free stream. This research is a notable effort to measure heat and mass transportation of the flow over an expanding cylinder with the imposition of the external flow. The system of complex partial differential equations is converted into highly nonlinear ordinary differential equations. The solution of the nonlinear system of equations is made with the numerical technique Runge‐Kutta fifth order just after the implementation of the shooting technique with suitable conditions. MATLAB solver bvp4c has solved the problem of the flow over stretching cylinder very efficiently and presents the facts in the form of the graphs and numerical values. The results claim that for the values of gamma parameter from 0.1 to 0.5 the rate of heat transfer increases by 14% on the other lambda with values from 0.1 to 0.3 the heat transfer rate declines 11%. For the growing values of Schmidt number from 1.0 to 5.0 the rate of mass transfer decreases by 85%. The rate of heat transfer has fallen by 86% for the improving values of the Prandtl number from 1, 2, and 3. [ABSTRACT FROM AUTHOR]

This paper focuses on an efficient computational method for solving the singularly perturbed Burger-Huxley equations. The difficulties encountered in solving this problem come from the nonlinearity term. The quasilinearization technique linearizes the nonlinear term in the differential equation. The finite difference approximation is formulated to approximate the derivatives in the differential equations and then accelerate its rate of convergence to improve the accuracy of the solution. The stability and consistency analysis were investigated to guarantee the convergence analysis of the formulated method. Numerical examples are considered for numerical illustrations. Numerical experiments were conducted to sustain the theoretical results and to show that the proposed method produces a more correct solution than some surviving methods in the literature. [ABSTRACT FROM AUTHOR]

The present work is focused on the stagnation point flow over a Jeffery fluid flow through a stretching/shrinking sheet. The nanoparticles namely Cu are considered by exploiting water-based fluid. The governing nonlinear system of partial differential equations is converted into a system of highly nonlinear ordinary differential equations via a similarity transformation. Therefore, it is quite significant to assimilate the analytical extension of heat transfer and mass transfer fluid in the presence of porous under the influence of slip velocity. The analytical solution available for nondimensional momentum, heat transfer, and concentration profiles across the boundary layer is examination is done by checking the various values of physical parameters viz., Prandtl number, Radiation parameter, stretching/shrinking parameter, and mass transpiration for the flow and heat transfer which are obtainable through graphs. The present work on nanofluids flowing through stretching/shrinking surfaces has numerous applications in biomedicine, solar energy, cooling, nuclear system cooling, etc. [ABSTRACT FROM AUTHOR]

The present study focuses on the MHD Radiative Casson fluid flow with the effect of triple diffusivity over a vertical porous wall along with convective boundary conditions. The governing equations for detecting the nature of the fluid under the influence of solutal diffusivity and thermal conductivity in triple diffusive boundary layer flow are derived. Non-linear partial differential equations are reduced to ordinary differential equations via similarity transformation. The BVP4C method in MATLAB software is then used to solve them. The outcomes of several physical dimensionless parameters like permeability, convective parameter, buoyancy ratio parameter, Casson parameter and chemical reaction parameter with source/sink impacts established by graphics. Also, the impression of the local skin friction coefficient, Nusselt number, and local Sherwood number are presented through the tables. [ABSTRACT FROM AUTHOR]

The DTM-Pade approximation is used in the current work to analyze the thermal behavior and thermal stresses of an annular fin while accounting for temperature-dependent thermal conductivity and internal heat generation. The energy problem is converted into a nonlinear ordinary differential equation (ODE) using non-dimensional parameters, and the DTM-Pade approximation is then utilized to provide an approximate analytical solution. The impacts of various settings on the temperature field are also graphically analyzed. It has been found that increasing the heat generation parameter causes the temperature distribution to improve. The growing thermo-geometric parameter values lead to an improvement in fin efficiency. [ABSTRACT FROM AUTHOR]

In this study, a computational analysis has been made for a steady, incompressible, MHD flow of the SWCNT-blood and the MWCNT-blood nanofluids past a non-linear stretching sheet of variable thickness. Darcy porous medium has been considered for fluid flow. Consequences of homogeneous-heterogeneous chemical reactions on heat and mass transfer along with Joule heating and viscous dissipation have been explored. Appropriate similarity transformations have been applied to convert the governing equations of nanofluid flow into non-linear ordinary differential equations. Galerkin finite element scheme has been used to examine the resulting system with corresponding boundary conditions. Numerical solutions of equations are demonstrated via graphs for various physical parameters, and these graphs have also been analyzed. Comparison of velocity, thermal, and concentration profiles of SWCNT-blood and MWCNT-blood nanofluids has been established. It is concluded that flow profiles of MWCNT-blood nanofluid dominate. Concentration profile declines with homogeneous and heterogeneous reaction parameters whereas increases with the Schimdt number for both nanofluids. Then, a comparison between present and the existing results was made, and they are in good agreement. After that, several physical quantities such as the local Nusselt number, skin friction coefficient, and concentration rate are exhibited in a table. [ABSTRACT FROM AUTHOR]

This study considers in depth the flow of the boundary layer of an incompressible, viscous, and steady Casson fluid through an expanding surface when thermal radiation, heat source, Soret, and Dufour effects are present. Using suitable similarity transformations, the governing non-linear partial differential equations are converted into coupled ordinary differential equations and then estimated using the MATLAB software bvp4c using a shooting process. Variable values of the parameters employed in the current inquiry are offered, together with solutions for the parameters of momentum, temperature, concentration, coefficient of local skin friction, and local Nusselt number. It is observed that the Casson fluid parameter improves fluid velocity and decreases the temperature. Also, it is found that thermal radiation and the presence of a heat source/sink increase the temperature of the fluid. We compared the present investigation with previously published papers and found them harmonious. [ABSTRACT FROM AUTHOR]

In this work, the role of asymmetry in the beam profile of finite Airy-Gaussian (AiG) laser is presented to explore its propagation dynamics in collisionless magnetized plasma. Emphasis is on analyzing the self-trapping condition for both extraordinary (E) and ordinary (O) modes of propagation of AiG laser beam. Theoretical formulation is based on obtaining nonlinear coupled differential equations for beam-width parameters in both transverse dimensions of the beam with the help of parabolic equation approach under WKB and paraxial approximations. It is found that asymmetry in the modulation parameters significantly affects self-focusing of finite AiG beam. Moreover, the same initial intensity parameter for simultaneous self-trapping of both modes of propagation is noteworthy. [ABSTRACT FROM AUTHOR]

The aim of this text is to elucidate the oscillating patterns (see C. Cheverry [Res. Rep. Math. (2018)]) which are generated in a toroidal plasma by a strong external magnetic field and a nonzero electric field. It is also to justify and then study new modulation equations which are valid for longer times than before. Oscillating coherent structures are induced by the collective motions of charged particles which satisfy a system of ODEs implying a large parameter, the gyrofrequency \varepsilon ^{-1} \gg 1. By exploiting the properties of underlying integrable systems, we can complement the KAM picture (see G. Benettin and P. Sempio [Nonlinearity 7 (1994), pp. 281–303]; M. Braun [SIAM Rev. 23 (1981), pp. 61–93]) and go beyond the classical results about gyrokinetics (see M. Bostan [Multiscale Model. Simul. 8 (2010), pp. 1923–1957]; A. J. Brizard and T. S. Hahm [Rev. Modern Phys. 79 (2007), pp. 421–468]). The purely magnetic situation was addressed by C. Cheverry [Comm. Math. Phys. 338 (2015), pp. 641–703; J. Differential Equations 262 (2017), pp. 2987–3033]. We are concerned here with the numerous additional difficulties due to the influence of a nonzero electric field. [ABSTRACT FROM AUTHOR]

In this paper, we present a converse Lyapunov theorem for the new notion of global generalized practical uniform h-stability of nonlinear systems of differential equations. We derive some sufficient conditions which guarantee the global generalized practical uniform h-stability of time-varying perturbed systems. In addition, these results are used to study the practical h-stability of models of infectious diseases and vaccination. [ABSTRACT FROM AUTHOR]

In this article, we discuss the existence of extremal solutions for a class of nonlinear sequential δ-Caputo fractional differential equations involving nonlinear boundary conditions. Our results are founded on advanced functional analysis methods. To be more specific, we use the monotone iterative approach in conjunction with the upper and lower solution method to create adequate requirements for the existence of extremal solutions. As an application, we give an example to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, we deal with initial value problems for coupled systems of nonlinear fractional differential equations, subject to coupled nonlocal initial and impulsive conditions on the half line. Global existence-uniqueness results are obtained under weak conditions allowing the reaction part of the problem to increase indefinitely with time. Our approach relies mainly to some fixed point theorem of Perov's type in generalized gauge spaces. The obtained results improve, generalize and complement many existing results in the literature. An example illustrating our main finding is also given. [ABSTRACT FROM AUTHOR]

The analysis of plane stress problems has long been a topic of interest in linear elasticity. The corresponding problem for non-linearly elastic materials is considered here within the context of homogeneous incompressible isotropic elasticity. It is shown that when the problem is posed in terms of the Cauchy stress, a semi-inverse approach must be employed to obtain the displacement of a typical particle. If however the general plane stress problem is formulated in terms of the Piola-Kirchhoff stress, the deformation of a particle requires the solution of a non-linear partial differential equation for both simple tension and simple shear, the trivial solution of which yields a homogeneous deformation. It is also shown that the general plane stress problem can be solved for the special case of the neo-Hookean material. [ABSTRACT FROM AUTHOR]

In this work, we explore the behavior of a classical RLC resonance-based bandpass filter, which includes two resistors (one of which is associated with a non-ideal inductor), when either of these resistors is self voltage-controlled. In particular, self-feedback control is achieved by using the voltage developed across the inductor or the capacitor to dynamically change the value of the controlled resistor. This results in a multiplication-type non-linearity, which transforms the linear filter into a non-linear filter described by a set of non-linear differential equations. When gradually increasing the strength of the non-linearity, a notch-like behavior is observed at twice the resonance frequency. However, the non-linear filter can lose its stability with excessive feedback. Simulations and experimental results are provided to support the theory. [ABSTRACT FROM AUTHOR]

This paper presents an analytical investigation of the solutions to a control-volume model for liquid films flowing down a vertical fibre. The evolution of the free surface is governed by a coupled system of degenerate nonlinear partial differential equations, which describe the fluid film's radius and axial velocity. We demonstrate the existence of weak solutions to this coupled system by applying a priori estimates derived from energy-entropy functionals. Additionally, we establish the existence of traveling wave solutions for the system. To illustrate our analytical findings, we present numerical studies that showcase the dynamic solutions of the partial differential equations as well as the traveling wave solutions. [ABSTRACT FROM AUTHOR]

Mathematical models and numerical simulations are necessary to understand the dynamical behaviors of complex systems. The aim of this work is to investigate closed-form solutions for the ball–plate problem considering a system derived from an optimal control problem for ball–plate dynamics. The nonlinear properties of ball and plate control system are presented in this work. To semi-analytically solve this system, we explored a second-order nonlinear differential equation. Consequently, we obtained the approximate closed-form solutions by the Optimal Parametric Iteration Method (OPIM) using only one iteration. A comparison between the analytical and corresponding numerical procedures reflects the advantages of the first one. The accordance between the obtained results and the numerical ones highlights that the procedure used is accurate, effective, and good to implement in applications such as sliding mode control to the ball-and-plate problem. [ABSTRACT FROM AUTHOR]

This article studied the new traveling wave solutions of the cascaded model with higherorder dispersion effects combined with the effects of spatiotemporal dispersion and multiplicative white noise. In the process of exploring traveling wave solutions, a clever combination of the polynomial complete discriminant system was used to discover more forms of traveling wave solutions for this equation. In order to better observe and analyze the propagation characteristics of traveling wave solutions, we used Maple and Matlab software to provide two-dimensional and three-dimensional visualization displays of the equation solutions. Meanwhile, we also analyzed the internal mechanism of nonlinear partial differential equations using planar dynamical systems. The research results indicated that there are differences in the results of different forms of soliton solutions affected by external random factors, which provided more beneficial references for people to better understand the cascaded model with higher-order dispersion effects combined with the effects of spatiotemporal dispersion and multiplicative white noise, and helped people to more comprehensively understand the propagation characteristics of optical solitons. The solution method in this article was also applicable to the study of other nonlinear partial differential equations. [ABSTRACT FROM AUTHOR]

The time-fractional partial differential equations were solved by the fractional natural transform decomposition method. Fractional derivatives are Caputo sense. The Fornberg-Whitham equation is a generalization of the Korteweg-de Vries (KdV) equation, which describes the propagation of long waves in shallow water. It includes higher-order dispersion terms, making it applicable to a wider range of dispersive media the fractional natural transform decomposition method (FNTDM) was also used to examine applications, and the solutions obtained by this method have been compared to those obtained by the variational iteration method, fractional variational iteration method, and homotopy perturbation method. In addition, the MAPLE package drew graphs of the solutions of nonlinear time-fractional partial differential equations, taking into account physics. The method described in this study exhibited a notable degree of computational precision and straightforwardness when used to the analysis and resolution of intricate phenomena pertaining to fractional nonlinear partial differential equations within the domains of science and technology. [ABSTRACT FROM AUTHOR]

This study examines the effect of heat production and radiation absorption on the magnetohydrodynamic Casson fluid flow at the stagnation point in a porous medium. We convert the group of fluid flow equations, which are non-linear partial differential equations with suitable boundary constraints, into a set of non-linear ordinary differential equations using similarity transformations. The homotopy analysis method (HAM) solves the converted system of ordinary differential equations. We draw graphs for numerous values of non-dimensional parameters and tables of surface drag force, rates of heat transfer, and mass transfer to analyze the relationship between velocity field, temperature field, concentration field, and other essential parameters involved in the study. We have proven that the Dufour number, radiation parameter, and heat generation parameter elevate the fluid temperature, whereas the magnetic parameter lowers it. The Casson fluid parameter, buoyancy force parameter, and mixed convection parameter all promote fluid movement throughout the flow field. The presented tabular data allows us to see the trend of heat and mass transfer rates, as well as drag force rates, against important parameters, enhancing our understanding of these rates. [ABSTRACT FROM AUTHOR]

This work considers the existence of solutions of the heteroclinic type in nonlinear second-order differential equations with ϕ -Laplacians, incorporating generalized impulsive conditions on the real line. For the construction of the results, it was only imposed that ϕ be a homeomorphism, using Schauder's fixed-point theorem, coupled with concepts of L 1 -Carathéodory sequences and functions along with impulsive points equiconvergence and equiconvergence at infinity. Finally, a practical part illustrates the main theorem and a possible application to bird population growth. [ABSTRACT FROM AUTHOR]

In this paper, the effect of the rotation variable and other variables on the peristaltic flow of Sutterby fluid in an Inclined asymmetric channel containing a porous medium with heat transfer is examined. In the presence of rotation, mathematical modeling is developed using constitutive equations based on the model of Sutterby fluid. In flow analysis, assumptions such as long wavelength approximation and low Reynolds number are used. The resulting nonlinear ordinary differential equation was analytically solved using the perturbation method. The effects of the Grashof number, the Hartmann number, the Reynold number, the Froude number, the Hall parameter, the Darcy number, the magnetic field, the Sutterby fluid parameter, and heat transfer analysis on the stream function, and the pressure gradient are analyzed graphically. Utilizing MATHEMATICA software, numerical results were computed. It is discovered that the size of boluses decreases as some parameters increase, whereas the pressure gradient is directly proportional to the majority of parameters. [ABSTRACT FROM AUTHOR]

Tuned liquid column damper (TLCD) is a well-known liquid damper designed to absorb the vibration of structures used in many applications, such as high-story buildings, wind turbines, and offshore platforms, requiring an accurate mathematical determination of the liquid level to model the TLCD structure system motion. The mathematical model of a TLCD is a nonlinear ordinary differential equation, unlike the structure, due to the term containing a viscous damping coefficient, and cannot be solved analytically. In this study, the fluid behavior of a TLCD without an orifice, directly connected to a shaking table under harmonic excitation, was investigated experimentally and a new linearization coefficient was proposed to be used in the mathematical model. First, the nonlinear mathematical model was transformed to a nondimensional form to better analyze the parameter relations, focusing on the steady-state amplitude of the liquid level during the harmonic excitation. The experimental data were then processed using the fourth-order Runge–Kutta method, and a correlation to calculate the viscous damping coefficient was proposed in the dimensionless form. Accordingly, a novel empirical model was proposed for the dimensionless steady-state amplitude of the liquid level using this correlation. Finally, with the help of the proposed correlation and the empirical model, an original linearization coefficient was introduced which does not need experimental data. The nonlinear mathematical model was linearized by using the developed linearization coefficient and solved analytically using the Laplace transform method. The study presents a generalized method for the analytical determination of the liquid level in a non-orifice TLCD under harmonic excitation, using a correlation and an empirical model proposed for the first time in this study, providing a novel and simple solution to be used in the examination of various TLCD structure systems. [ABSTRACT FROM AUTHOR]

Computation of the solution of the nonlinear Caputo fractional differential equation is essential for using q , which is the order of the derivative, as a parameter. The value of q can be determined to enhance the mathematical model in question using the data. The numerical methods available in the literature provide only the local existence of the solution. However, the interval of existence is known and guaranteed by the natural upper and lower solutions of the nonlinear differential equations. In this work, we develop monotone iterates, together with lower and upper solutions that converge uniformly, monotonically, and quadratically to the unique solution of the Caputo nonlinear fractional differential equation over its entire interval of existence. The nonlinear function is assumed to be the sum of convex and concave functions. The method is referred to as the generalized quasilinearization method. We provide a Caputo fractional logistic equation as an example whose interval of existence is [ 0 , ∞). [ABSTRACT FROM AUTHOR]

We derive the prolongation formula of the one-parameter Lie point transformations to the Hilfer fractional derivative and show that the existing prolongation formula for the Riemann Liouville and Caputo fractional derivatives are special cases of the proposed formula, corresponding to the type parameter γ = 0 and γ = 1 , respectively. The applicability of the proposed formula is demonstrated by deriving the Lie point symmetries of the time-fractional heat equation, the fractional Burgers equation, and the fractional KdV equation in Hilfer's sense. We use the obtained Lie point symmetries to find the similarity variables and transformations. Using the similarity transformations, we show that each is converted into a nonlinear fractional ordinary differential equation with a new independent variable. The fractional derivative in the reduced equation can be either the Hilfer-type modification of the Erdélyi Kober fractional derivative or the Hilfer fractional derivative itself. We demonstrate that the exact solution of the time-fractional differential equation in the Hilfer sense can be reduced to the exact solutions of the corresponding time-fractional differential equations in the Riemann–Liouville and Caputo senses by setting the type parameter to γ = 0 and γ = 1 , respectively. [ABSTRACT FROM AUTHOR]

A novel framework is proposed in this research based on multilevel method to solve the optimal control problem. In recent dacades, the mathematical theory of optimal control has rapidly developed into an important and separate field of applied mathematics. The solution of nonlinear partial differential equations is considerably difficult, and the theory of their optimal control is still an open field in many respects. These optimization problems have found diverse applications in various sciences including electrical engineering, mechanical engineering, and aerospace. Current methods for solving this class of optimal control problems usually fall into two classes: discrete-then-optimization or optimization-then-discrete approaches. The proposed approach, however, does not require discretization as it involves rewriting the optimal control problem as a multi-objective optimization problem followed by its solution with a feedforward single-layer artificial neural network based on learning through by the multi-level Levenberg–Marquardt method. Moreover, the convergence of the approach was discussed and some numerical results are presented. [ABSTRACT FROM AUTHOR]

Pyrazinamide (PZA) is a key drug in the treatment of Mycobacterium tuberculosis. Although not completely understood yet, the bactericidal mechanism of PZA starts with its diffusion into the cell and subsequent conversion into pyrazinoic acid (POA) after the hydrolysis of ammonia group. This leads to the acidification cycle, which involves: (1) POA extrusion into the extracellular environment, (2) reentry of protonated POA, and (3) release of a proton into the cytoplasm, resulting in acidification of the cytoplasm and accumulation of intracellular POA. To better understand this process, we developed a system of coupled non-linear differential equations, which successfully recapitulates the kinetics of PZA/POA observed in M. tuberculosis. The parametric space was explored, assessing the impact of different PZA and pH concentrations and variations in the kinetic parameters, finding scenarios of PZA susceptibility and resistance. Furthermore, our predictions show that the acidification cycle alone is not enough to result in significant intracellular accumulation of POA in experimental time scales when compared to other neutral pH scenarios. Thus, revealing the need of novel hypotheses and experimental evidence to determine the missing mechanisms that may explain the pH-dependent intracellular accumulation of POA and their subsequent effects. [ABSTRACT FROM AUTHOR]

This study seeks to explore the integrable dynamics of induced curves through the utilization of the complex-coupled Kuralay system. The importance of the coupled complex Kuralay equation lies in its role as a fundamental model that contributes to the understanding of intricate physical and mathematical concepts, making it a valuable tool in scientific research and applications. The soliton solutions originating from the Kuralay equations are believed to encapsulate cutting-edge research in various important domains such as optical fibers, nonlinear optics, and ferromagnetic materials. Analytical techniques are employed to derive traveling wave solutions for this model, given that the Cauchy problem cannot be resolved using the inverse scattering transform. In the quest for solitary wave solutions, the extended modified auxiliary equation mapping (EMAEM) method is employed. We derive several novel families of precise traveling wave solutions, encompassing trigonometric, hyperbolic, and exponential forms. Moreover, the planar dynamical system of the concerned equation is created, all probable phase portraits are given, and sensitive inspection is applied to check the sensitivity of the considered equation. Furthermore, after adding a perturbed term, chaotic and quasi-periodic behaviors have been observed for different values of parameters, and multistability is reported at the end. Numerical simulations of the solutions are incorporated alongside the analytical results to enhance comprehension of the dynamic characteristics of the solutions obtained. This study’s outcomes can offer valuable insights for addressing other nonlinear partial differential equations (NLPDEs). The soliton solutions obtained in this study offer important insights into the intricate nonlinear equation being examined. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of nonlinear differential equations with multiple distributed delays. We reduced the problem of search for periodic solutions of the distributed delay differential equations to that of finding zeros of an operator on some Banach space. Utilizing the equivariant degree theory, we obtain some sufficient conditions to guarantee the existence of multiplicity of periodic solutions. Also, we give some information about the symmetry of the periodic solutions. At last, we give an example to illustrate our result. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a rehab–recovery–relapse cycle model to study the interaction between healthy individuals, individuals that might develop substance use disorders (SUD), which for simplicity here we will call susceptible individuals, individuals with SUD, and individuals recovered in a rehab community. We measure the community’s health through a system defined by four non-linear ordinary differential equations in which we include a rehab community. After computing the SUD-free and coexistence equilibrium points, we find the feasibility and stability conditions of an equivalent and reduced system obtained by nondimensionalizing the equations of the original model. We have numerically investigated the importance of the parameter values on the model’s outcome via a one- and two-strain parameter analysis. From the numerical results, we observe that an efficient rehab community can be characterized by (i) a high recovery rate (the individuals stay in that community as little time as possible) and (ii) permanent rehabilitation being preferred at the expense of a low recovery rate. [ABSTRACT FROM AUTHOR]

This article discusses the use of non-smooth variational methods in a wide range of applications, particularly in fields that involve partial differential equations. The authors focus on nonlinear problems in solid and fluid mechanics, as well as inverse and optimal control problems. They aim to develop non-standard analysis and effective numerical methods for solving these problems. The article also highlights the contributions of various authors from different countries and discusses the physical applications of non-smooth variational methods in areas such as elastic and inelastic solids, heat transfer, and fracture mechanics. [Extracted from the article]

The process of convective thermal management in the radiative flow of Oldroyd-B fluid over the stretching cylinder is investigated in this paper. In engineering and applied sciences, the viewpoint of fluid flow due to vertically stretched cylinder covers both theoretical and practical significances. Nonlinear ordinary differential equations are first transformed from their corresponding partial differential equations, then rebuilt using sufficient transformations to produce a system of nondimensional equations, using a proper similarity technique, which is then calculated. The MATLAB built-in technique BVP4C, a standard numerical process, is used to solve the regulating flow equations. A comprehensive graphical depiction was used for characteristics like unsteadiness and curvature, Prandtl number, and thermal diffusivity. The relevant flow characteristics of the governing problem are depicted in graphs. [ABSTRACT FROM AUTHOR]

Research into hybrid nanofluid flow over a disk surface is growing due to numerous processes in marine or gas turbines, rotating disk reactors for biofuels generation, cooling of spinning equipment components and other industrial applications. The purpose of this research study is to examine the entropy generation for the transient thermocapillary flow of hybrid nanoliquid thin films across a disk surface with different shapes of nanoparticles. To analyze the nanomaterial, the Tiwari–Das hybrid nanofluid flow model is established, and in doing so, Prandtl's boundary layer theory is incorporated into this model. The energy equation considers the impacts of thermal radiation, two different kinds of heat sources — namely an exponentially space-dependent heat source and a linear thermal heat source — and viscous dissipation. The nonlinear partial differential equations, which explain the flow processes, are transformed into the nondimensional ordinary differential equations (ODEs). Once the ODEs have been constructed, they are next solved using the bvp4c technique. In addition, the surface drag force and the rate of heat transfer are both evaluated as functions of the shape factor of the nanoparticles that are disseminated in the base fluid. The influence of significant parameters on the flow fields is depicted graphically, and adequate physical explanations are provided for each representation. One of the important findings of this study indicates that the largest amount of heat transfer can be accomplished at the disk surface by employing nanoparticles with blade shape, while this physical quantity is the least when nanoparticles with spherical shapes are used. The temperature profile grows as the thermal and exponential heat sources increase. Entropy generation improves with the increase in either the magnetic parameter or the Brinkman number. [ABSTRACT FROM AUTHOR]

The utilization of a ternary hybrid nanofluid, a recent development in the realm of nanofluids, can result in improved heat transfer. In the ongoing study, a ternary hybrid nanofluid flow is utilized, and it is carried out atop a porous spinning disc, which is exposed to a magnetic field, heat generation, thermal radiation, and Casson fluid. In this study, Blood/Water are taken as base fluids and (Ag–Au–Al2O3) are considered as a ternary hybrid nanoparticle. Nanoparticles made of gold and silver are put to use in a vast number of industries and fields, including nanotechnology and medicine. As a result of surface effects and quantum effects, these precious metals exhibit unique features in nanoform that play a vital role in with optical, magnetic, chemical, and mechanical behavior. Al2O3 unique optical, physical and biochemical qualities make it worthwhile for numerous uses, including nanophotonic, catalysis and the fabrication of high-energy composites. A set of relevant similarity transformations is used to generate non-dimensional forms of controlling paired nonlinear Partial Differential Equations (PDEs). MATLAB is used to perform a numerical solution with ODE45. In addition, the velocity outline decreases, and the temperature profile increases slightly before decreasing over a revolving disk when the values of magnetic parameters are increased. The distribution and radiant heat components heat up as the level gets higher. Aim and objectives of the study: The aim of this study is to analyze the Comprehensive Scrutinization of Ternary hybrid and Casson flow in a conducting porous rotating disk with internal heating. The primary objective of this analysis is to increase awareness of the impending energy crisis among those working in the industrial and technological sectors. Ternary nanoparticles (NPs) have a wide range of applications, which lends credence to the developed model. For example, Al2O3 can be used in a variety of ways that benefit society, it is used in water purification to remove water from the gas streams and extend people's lives. [ABSTRACT FROM AUTHOR]

This study investigates analytically the stability analysis of a nonlinear convective boundary layer (BL) flow of the nanofluids GO-EG and GO-W on a shrinking surface in the presence of a magnetic field and viscous dissipation. We provide a second-order nonlinear ordinary differential equation (NODE) for temperature distribution (TD) and a third-order NODE for velocity profile (VP) coupling based on the thermophysical characteristics of the base fluid and nanofluid as well as similarity transformations (STs) in the fundamental governing equations of momentum and energy. The analytical method HAM is used to answer the equations that have been gathered. In many different industries, such as manufacturing, automotive, microelectronics, and defense, cooling of various types of equipment and devices has very important challenges. To fulfill the demands of the engineering and industrial industries, it is hoped that this issue would significantly improve the heat transformation ratio. [ABSTRACT FROM AUTHOR]

Recently, the area devoted to fractional calculus has given much attention by researchers. The reason behind such huge attention is the significant applications of the mentioned area in various disciplines. Different problems of real world processes have been investigated by using the concepts of fractional calculus and important and applicable outcomes were obtained. Because, there has been a lot of interest in fractional differential equations. It is brought on by both the extensive development of fractional calculus theory and its applications. The use of linear and quadratic perturbations of nonlinear differential equations in mathematical models of a variety of real-world problems has received a lot of interest. Therefore, motivated by the mentioned importance, this research work is devoted to analyze in detailed, a class of fractal hybrid fractional differential equation under Atangana- Baleanu- Caputo ABC derivative. The qualitative theory of the problem is examined by using tools of non-linear functional analysis. The Ulam-Hyer's (U-H) type stability criteria is also applied to the consider problem. Further, the numerical solution of the model is developed by using powerful numerical technique. Lastly, the Wazewska-Czyzewska and Lasota Model, a well-known biological model, verifies the results. Several graphical representations by using different fractals fractional orders values are presented. The detailed discussion and explanations are given at the end. [ABSTRACT FROM AUTHOR]

The utilization of nonlinear differential equations has resulted in remarkable progress across various scientific domains, including physics, biology, ecology, and quantum mechanics. Nonetheless, obtaining multiple solutions for nonlinear differential equations can pose considerable challenges, particularly when it is difficult to find suitable initial guesses. To address this issue, we propose a pioneering approach known as the Companion-Based Multilevel Finite Element Method (CBMFEM). This novel technique efficiently and accurately generates multiple initial guesses for solving nonlinear elliptic semi-linear equations containing polynomial nonlinear terms through the use of finite element methods with conforming elements. As a theoretical foundation of CBMFEM, we present an appropriate and new concept of the isolated solution to the nonlinear elliptic equations with multiple solutions. The newly introduced concept is used to establish the inf-sup condition for the linearized equation around the isolated solution. Furthermore, it is crucially used to derive a theoretical error analysis of finite element methods for nonlinear elliptic equations with multiple solutions. A number of numerical results obtained using CBMFEM are then presented and compared with a traditional method. These not only show the CBMFEM's superiority, but also support our theoretical analysis. Additionally, these results showcase the effectiveness and potential of our proposed method in tackling the challenges associated with multiple solutions in nonlinear differential equations with different types of boundary conditions. [ABSTRACT FROM AUTHOR]

This paper presents a novel approach to studying divorce dynamics and elimination strategies using nonlinear differential equations. A mathematical model is formulated to capture the key factors influencing divorce rates. The model undergoes a rigorous theoretical analysis, including parameter estimation, solution existence/uniqueness, positivity, boundedness, and invariant regions. A qualitative analysis explores equilibria, stability conditions, and a sensitivity analysis. Numerical simulations and discussions are presented to validate the model and shed light on divorce dynamics. Finally, conclusions and future research directions are outlined. This work offers valuable insights for understanding and potentially mitigating divorce rates through targeted interventions. [ABSTRACT FROM AUTHOR]

An unsteady two-dimensional nanofluid flow across a permeable stretched sheet is examined in terms of thermodynamic analysis. Water is utilized as the basic fluid, together with four distinct types of nanoparticles: copper (Cu), copper oxide (CuO), aluminum oxide (Al2O3), and titanium dioxide (TiO2). An external source is used to describe the heat transfer phenomena. The effects of heat generation and absorption are also taken into account. The nonlinear partial differential equations are transformed into a system of nonlinear ordinary differential equations via a similarity transformation. The spectral quasi-linearization approach is then used to numerically integrate the altered equations. Except when otherwise noted, the findings are provided for copper nanoparticle situations. A visual and detailed explanation of the effects of several physical factors on the profiles of dimensionless velocity, temperature, average entropy generation function, skin friction, and the Nusselt number are discussed. As exceptional circumstances of the current findings, there is a strong agreement between the current conclusion and the findings of the other researchers. The average entropy generation number, temperature, and velocity profiles are shown to be strongly influenced by regulating factors. It is concluded that the average entropy generation decreased in the presence of a temperature- and space-dependent heat source/sink, but it improves with enhanced viscosity parameter. The significance of the present study has frequent applications in the major area of heat transfer enhancement in renewable energy systems, industrial thermal management, and also materials processing. [ABSTRACT FROM AUTHOR]

The present investigation concerns the hydro-thermal characteristics of two different types of hybrid nanofluid flow, namely MoS2-SiO2/water hybrid nanofluid and MoS2-SiO2-GO/water ternary hybrid nanofluid flow toward an inclined cylinder in the presence of heat generation/absorption and viscous dissipation. Emerging problem occurs as nonlinear partial differential equations (PDEs) included conservation laws of mass, momentum and heat transportation. The relatable problem can be changed into the set of ordinary differential equations (ODEs) using similarity transformations. The Runge-Kutta-Fehlberg (RKF) technique via shooting scheme is adopted to tackle obtained set of ODEs. The influence of significant physical factors on these hybrid nanofluids such as suction/injection, magnetic field, viscous dissipation, and thermal radiation is discussed and depicted by graphs. Also, tabulated results are provided for the skin friction coefficient and rate of heat transfer of both hybrid and ternary hybrid nanofluids. The outcomes describe that the thermal profiles are higher for the ternary hybrid nanofluid than hybrid nanofluid for increasing values of thermal radiation parameter and Eckert number. Also, the velocity profiles of both hybrid nanofluids experience a reduction when higher values of suction/injection parameter are applied. Furthermore, changing the value of the heat generation/absorption parameter from 2.0 to −2.0 leads to 14.74% increment in the heat transfer rate in the hybrid nanofluid, whereas the ternary hybrid nanofluid experience a more significant 28.46% enhancement in the heat transfer rate. [ABSTRACT FROM AUTHOR]

In recent years, there has been a growing trend in modern engineering systems to integrate electromagnetic features with bio-convection phenomena. This integration has found applications in various fields such as materials processing, robotics, nano-sensors, and computational biology. Inspired by these advancements, a computational simulation is presented to study the steady bio-convection thermo-solutal flow of a Casson nano-fluid, which is a visco-plastic fluid. The simulation takes into account factors such as thermal radiation, heat generation, Soret and Dufour effects, chemical reaction, and convective boundary conditions. It also considers the presence of gyrotactic microorganisms. To model the Casson nano-fluid, a revised Casson model is implemented using the Buongiorno nanoscale formulation. The boundary layer concept is used to derive the fundamental conservation partial differential equations for mass, momentum, energy, solute concentration, and microorganism number density, along with appropriate boundary conditions. The model is then transformed into a system of coupled nonlinear ordinary differential equations using suitable transformations. Numerical solutions are obtained using the built-in method in a Mathematica. The results are presented graphically to provide a visual representation of the flow parameters. The main findings are summarized in the conclusion section. [ABSTRACT FROM AUTHOR]

Nonlinear time-fractional partial differential equations (NTFPDEs) play a great role in the mathematical modeling of real-world phenomena like traffic models, the design of earthquakes, fractional stochastic systems, diffusion processes, and control processing. Solving such problems is reasonably challenging, and the nonlinear part and fractional operator make them more problematic. Thus, developing suitable numerical methods is an active area of research. In this paper, we develop a new numerical method called Yang transform Adomian decomposition method (YTADM) by mixing the Yang transform and the Adomian decomposition method for solving NTFPDEs. The derivative of the problem is considered in sense of Caputo fractional order. The stability and convergence of the developed method are discussed in the Banach space sense. The effectiveness, validity, and practicability of the method are demonstrated by solving four examples of NTFPEs. The findings suggest that the proposed method gives a better solution than other compared numerical methods. Additionally, the proposed scheme achieves an accurate solution with a few numbers of iteration, and thus the method is suitable for handling a wide class of NTFPDEs arising in the application of nonlinear phenomena. [ABSTRACT FROM AUTHOR]

In this paper, the thermal behavior of cylindrical and conical pin fin structures operating in a convective–radiative environment has been examined. The fully wet porous fin structures are assumed to be mounted with a stretching/shrinking mechanism similar to a conveyer belt. The conical pin fin has been comparatively analyzed with the cylindrical one for three distinct cases of stretching, stagnant, and shrinking. Further, the Darcy model has been employed to analyze the fluid–solid interactions. The differential transformation method (DTM) has been used to solve the resulting second-order nonlinear ordinary differential equations. On the thermal performance of conical and cylindrical pin fin structures, the effects of the stretching/shrinking parameter, Peclet number, wet porous parameter, convective parameter, and radiative parameter have been established graphically. The conical pin fin is found to aid the heat transmission process better than the cylindrical one. The work is also helpful in the realm of microelectronics, particularly in the development of micro-pin-fin structures. [ABSTRACT FROM AUTHOR]

We obtain a nonlinear second-order differential equation that describes forced vibrations in a milking machine. Approximate analytical solutions of the obtained differential equations are given. [ABSTRACT FROM AUTHOR]

AbstractThe free convective, unsteady, two-dimensional (2D) magnetohydrodynamic (MHD) Darcy–Forchheimer Casson fluid flow via a vertical stretching sheet containing gyrotactic microorganisms was the novel aspect of this investigation. We rewrote the collection of partial differential equations (PDEs) as nonlinear ordinary differential equations (ODEs) by considering an appropriate similarity variable. We solved the overhaul of nonlinear ODEs using MATLAB’s bvp4c approach. This study rigorously investigates the significance of gyrotactic microorganisms for Casson fluid motility. Additionally, this work investigates the impact of various parameters, including magnetic parameters, thermophoresis, Brownian motion, and radiation parameters, on temperature, motile microorganisms, velocity, and concentration profiles. We use graphs to illustrate the outcomes of various fluid flow parameters. Numerical tables display the effects of different parameters on skin friction, Sherwood number, Nusselt number, and the quantity of motile microorganisms. The unsteady parameter causes a drop in the velocity, temperature, concentration, and the density profile of microorganisms. Additionally, the Casson parameter leads to a decrease in the flow profile, while simultaneously increasing the heat, concentration, and microorganism density profiles. The microorganism density profile decreases as the bioconvection Schmidt number, Peclet number, and motile microbe parameter increase. We deem the new results extremely satisfactory when compared to earlier research. The fields of environmental remediation, biomedical sciences, and engineering may find this examination useful. [ABSTRACT FROM AUTHOR]

A novel Haar scale-3 wavelet collocation technique is proposed in this study for dealing with a specific type of parabolic Buckmaster second-order nonlinear partial differential equation in a dispersive system and Chafee-Infante second-order non-linear partial differential equation (PDE) in a solitary system. Using Haar scale-3 (HSW-3) wavelets, the system approximates the space and time derivatives. To develop both an implicit and explicit analytical model for the dispersive and solitary system, the collocation approach is employed in conjunction with the discretization of space and time variables. We have examined the effectiveness, applicability, and veracity of the proposed computational approach using a variety of numerical problems with nonlinearity and numerous significant source terms. Additionally, the outcomes are graphically presented and organized. We achieved accuracy with the proposed methods even with a small selection of collocation locations. [ABSTRACT FROM AUTHOR]

We present an analysis of generalized Nash equilibrium problems in infinite-dimensional spaces with possibly non-convex objective functions of the players. Such settings arise, for instance, in games that involve nonlinear partial differential equation constraints. Due to non-convexity, we work with equilibrium concepts that build on first order optimality conditions, especially Quasi-Nash Equilibria (QNE), i.e. first-order optimality conditions for (Generalized) Nash Equilibria, and Variational Equilibria (VE), i.e. first-order optimality conditions for Normalized Nash Equilibria. We prove existence of these types of equilibria and study characterizations of them via regularized (and localized) Nikaido-Isoda merit functions. We also develop continuity and (continuous) differentiability results for these merit functions under quite weak assumptions, using a generalization of Danskin's theorem. They provide a theoretical foundation for, e.g. using globalized descent methods for computing QNE or VE. [ABSTRACT FROM AUTHOR]

The fractional nonlinear generalized (3 + 1) -dimensional Chaffee–Infante (FGCI) equation is a nonlinear evolution equation that governs the propagation of waves in dispersive media, exhibiting both dissipative and dispersive effects. It finds applications in various fields such as fluid mechanics, plasma physics, and nonlinear optics, where the interplay between nonlinearity, dispersion, and dissipation plays a crucial role in determining the dynamics of the system. This study aims to obtain analytical and numerical solutions for the FGCI equation, which is a generalization of the Chaffee–Infante equation. The analytical approach employed is the Khater II (Khat II) method, an efficient technique for constructing approximate analytical solutions to nonlinear fractional differential equations. Additionally, He's variational iteration (HVI) method is implemented as a numerical scheme to validate the accuracy of the analytical solutions obtained from the Khat II method. The results obtained through these methods demonstrate the effectiveness of the proposed techniques in solving the FGCI equation and provide valuable insights into the dynamics of wave propagation phenomena governed by the equation. The significance of this study lies in its contribution to the understanding of nonlinear wave propagation phenomena and the development of analytical and numerical techniques for solving fractional differential equations. The novelty of the research lies in the application of the Khat II method to the FGCI equation, which has not been previously explored, and the validation of the analytical solutions through the HVI method. The study is conducted in the field of applied mathematics, with a focus on nonlinear fractional differential equations and their applications in various physical systems. [ABSTRACT FROM AUTHOR]