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In the current study, we solve the nonlinear time fractional‐order enzyme kinematic system and the linear time fractional blood alcohol level using an effective method called the Adomian natural decomposition method (ANDM), which is a combination of the Adomian decomposition method (ADM) and the natural transform method (NTM). On the basis of the Banach fixed point theorem, we offer proofs for the existence and uniqueness theorems as they apply to nonlinear ordinary differential equations. Using the current method, approximate analytical solutions of nonlinear time fractional‐order enzyme kinematic system and exact solutions to the linear time fractional‐order blood alcohol level problem have been obtained. The outcomes indicate that the ANDM is highly efficient and useful. The method outlined in this paper is intended to be applied in subsequent work to handle comparable nonlinear problems connected to fractional calculus. [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]

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]

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 work applies the Aboodh transform with the homotopy perturbation method (HPM) to solve fractional differential equations. As the Aboodh transform is limited to linear equations only, HPM is an effective and dominant method for nonlinear differential equations to obtain approximate solutions. The role of NWSE is important in nonlinear systems that explain how stripes arise in two‐dimensional systems. The ATHPM solution has also been compared with LTDM and VIM methods, and it has been shown that ATHPM has more accuracy with a less absolute error than the exact solution, VIM, and LTDM. The final results perform very well with the exact solution. Maple is used to represent 3‐D surfaces and to find numerical values shown in tables. [ABSTRACT FROM AUTHOR]

This work delves into the behavior of cold bosonic atoms within a zigzag optical lattice which offers a rich platform for studying quantum many‐body physics and has potential applications in various fields including quantum simulation, quantum computing, and precision measurement. The study aims to derive exact solutions and explore qualitative properties through dynamical analysis. Exact solutions of the nonlinear differential equation model can provide insights into the behavior of bosonic atoms in complex optical lattice configurations, allowing researchers to simulate and study phenomena such as quantum phase transitions, many‐body localization, and exotic states of matter. The invariant analysis has been performed for the first time on the specified equation, yielding a reduced system of equations with forms that are easier to handle. Novel techniques are applied to extract solutions from the reduced system of equations obtained via invariant analysis. The results reveal a rich set of solutions, including various traveling wave solutions and doubly periodic functions in the form of Jacobian elliptic functions. Bifurcation analysis, conducted through phase portraits, provides insights into the system's long‐term behavior under different parameter values. This work contributes to a deeper understanding of the dynamics of cold bosonic atoms in optical lattices via 2D and 3D plots of obtained solutions depicting the change in the behavior of soliton solutions with fractional derivative parameter. [ABSTRACT FROM AUTHOR]

The current study is important from two perspectives. Firstly, in this article, we suggest a novel analytical technique for creating the exact solutions to nonlinear partial differential equations (NLPDEs). In order to study the dynamical behaviors of various wave phenomena, we can construct the several exact solutions in the form of Jacobi elliptic solutions, hyperbolic solutions, trigonometric solutions, and exponential solutions by using this method. Secondly, we consider a more generalized form of the (2+1)‐dimensional Korteweg–de Vries (KdV) and modified Korteweg‐de Vries (mKdV) equations that plays an important role in describing the shallow water. We successfully extract several soliton solutions for the examined equation. By choosing the appropriate parameters, some graphs of the discovered solutions have been represented in the figures. Every obtained solution has been demonstrated to satisfy the corresponding equation. The findings demonstrate that the method can be applied to solve a number of nonlinear evolution equations. The new solutions and the paper's findings could improve our understanding of how the waves move through shallow water and open up new research avenues. [ABSTRACT FROM AUTHOR]

Zika is a flavivirus that is transmitted to humans either through the bites of infected Aedes mosquitoes or through sexual transmission. Zika has been associated with congenital anomalies, like microcephalus. We developed and analyzed the fractional‐order Zika virus model in this paper, considering the vector transmission route with human influence. The model consists of four compartments: susceptible individuals are x1(t), infected individuals are x2(t), x3(t) shows susceptible mosquitos, and x4(t) shows the infected mosquitos. The fractional parameter is used to develop the system of complex nonlinear differential equations by using Caputo and Atangana–Baleanu derivative. The stability analysis as well as qualitative analysis of the fractional‐order model has been made and verify the non‐negative unique solution. Finally, numerical simulations of the model with Caputo and Atangana Baleanu are discussed to present the graphical results for different fractional‐order values as well as for the classical model. A comparison has been made to check the accuracy and effectiveness of the developed technique for our obtained results. This investigative research leads to the latest information sector included in the evolution of the Zika virus with the application of fractional analysis in population dynamics. [ABSTRACT FROM AUTHOR]

This paper is devoted to study the rotational periodic boundary value problem for a fractional‐order nonlinear differential equation. Applying topology‐degree theory and the Leray‐Schauder fixed‐point theorem, we prove the existence and uniqueness of solution for the fractional‐order differential system. Furthermore, the existence of solution for a nonlinear differential system with a multivalued perturbation term is investigated by using set‐valued theory and techniques of functional analysis. Two examples of applications are given at the end. [ABSTRACT FROM AUTHOR]

The problem of the existence and uniqueness of solutions of boundary value problems (BVPs) for a nonlinear fractional differential equation of order 2<α ≤ 3 is studied. The BVP is transformed into an integral equation and discussed by means of a fixed point problem for an integral operator. Conditions for the existence and uniqueness of a fixed point for the integral operator are derived via b‐comparison functions on complete b‐metric spaces. In addition, estimates for the convergence of the Picard iteration sequence are given. An estimate for the Green's function related with the problem is provided and employed in the proof of the existence and uniqueness theorem for the solution of the given problem. Illustrative examples are presented to support the theoretical results. [ABSTRACT FROM AUTHOR]

We introduce models for viscoelastic materials, both solids and fluids, based on logarithmic stresses to capture the elastic contribution to the material response. The matrix logarithm allows to link the measures of strain, that naturally belong to a multiplicative group of linear transformations, to stresses, that are additive elements of a linear space of tensors. As regards the viscous stresses, we simply assume a Newtonian constitutive law, but the presence of elasticity and plastic relaxation makes the materials non‐Newtonian. Our aim is to discuss the existence of weak solutions for the corresponding systems of partial differential equations in the nonlinear large‐deformation regime. The main difficulties arise in the analysis of the transport equations necessary to describe the evolution of tensorial measures of strain. For the solid model, we only need to consider the equation for the left Cauchy–Green tensor, while for the fluid model, we add an evolution equation for the elastically‐relaxed strain. Due to the tensorial nature of the fields, available techniques cannot be applied to the analysis of such transport equations. To cope with this, we introduce the notion of charted weak solution, based on non‐standard a priori estimates, that lead to a global‐in‐time existence of solutions for the viscoelastic models in the natural functional setting associated with the energy inequality. [ABSTRACT FROM AUTHOR]

In this study, the authors present a uniform algebraic trigonometric tension B‐spline‐based differential quadrature method combined with an optimized hybrid block method to numerically solve the Rosenau–KdV–RLW equation. The discrete mass and energy have been calculated, showing that they are conserved, thus indicating the efficiency and accuracy of the present approach. The method outperforms other methods and does not require linearization, allowing it to be directly implemented for solving nonlinear partial differential equations. [ABSTRACT FROM AUTHOR]

Stochastic differential equations (SDEs) are very useful in modeling many problems in biology, economic data, turbulence, and medicine. Fractional Brownian motion (fBm) and variable‐order fractional Brownian motion (vofBm) are suitable alternatives to standard Brownian motion (sBm) for describing and modeling many phenomena, since the increments of these processes are dependent of the past and for H>12$$ \mathcal{H}>\frac{1}{2} $$ these increments have the property of long‐term dependence. Classical mathematical techniques such as Ito's calculus do not work for stochastic computations on fBm and vofBm due to they are not semi‐Martingale for H(ξ)≠12$$ \mathcal{H}\left(\xi \right)\ne \frac{1}{2} $$. Therefore, solving these equations is much more difficult than solving SDEs with sBm. On the other hand, these equations do not have an analytical solution, so we have to use numerical methods to find their solution. In this paper, a computational approach based on hybrid of block‐pulse and parabolic functions (HBPFs) has been introduced for simulating vofBm and solving a modern class of SDEs. The mechanism of this approach is based on stochastic and fractional integration operational matrices, which transform the intended problem to a nonlinear system of algebraic equations. Thus, the complexity of solving the mentioned problem is reduced significantly. Also, convergence analysis of the expressed method has been theoretically examined. Finally, the accuracy and efficiency of the proposed algorithm have been experimentally investigated through some test problems and comparison of obtained results with results of previous papers. High accurate numerical results are obtained by using a small number of basic functions. Therefore, this method deals with smaller matrices and vectors, which is one of the most important advantage of our suggested method. Also, presenting an applicable procedure to construct vofBm is another innovation of this work. To gain this aim, at first, discretized sBm is generated via fundamental features of this process, and afterward, block‐pulse functions (BPFs) and HBPFs are utilized for simulating discretized vofBm. Finally, spline interpolation method has been employed to provide a continuous path of vofBm. [ABSTRACT FROM AUTHOR]

Numerical methods for solving differential equations often rely on the expansion of the approximate solution using basis functions. The choice of an appropriate basis function plays a crucial role in enhancing the accuracy of the solution. In this study, our aim is to develop algorithms that can identify an optimal basis function for any given differential equation. To achieve this, we explore fractional rational Jacobi functions as a versatile basis, incorporating hyperparameters related to rational mappings, Jacobi polynomial parameters, and fractional components. Our research develops hyperparameter optimization algorithms, including parallel grid search, parallel random search, Bayesian optimization, and parallel genetic algorithms. To evaluate the impact of each hyperparameter on the accuracy of the solution, we analyze two benchmark problems on a semi‐infinite domain: Volterra's population model and Kidder's equation. We achieve improved convergence and accuracy by judiciously constraining the ranges of the hyperparameters through a combination of random search and genetic algorithms. Notably, our findings demonstrate that the genetic algorithm consistently outperforms other approaches, yielding superior hyperparameter values that significantly enhance the quality of the solution, surpassing state‐of‐the‐art results. [ABSTRACT FROM AUTHOR]

This paper proposes two efficient approximation methods to solve high-dimensional fully nonlinear partial differential equations (NPDEs) and second-order backward stochastic differential equations (2BSDEs), where such high-dimensional fully NPDEs are extremely difficult to solve because the computational cost of standard approximation methods grows exponentially with the number of dimensions. Therefore, we consider the following methods to overcome this difficulty. For the merged fully NPDEs and 2BSDEs system, combined with the time forward discretization and ReLU function, we use multiscale deep learning fusion and convolutional neural network (CNN) techniques to obtain two numerical approximation schemes, respectively. Finally, three practical high-dimensional test problems involving Allen-Cahn, Black-Scholes-Barenblatt, and Hamilton-Jacobi-Bellman equations are given so that the first proposed method exhibits higher efficiency and accuracy than the existing method, while the second proposed method can extend the dimensionality of the completely NPDEs-2BSDEs system over 400 dimensions, from which the numerical results highlight the effectiveness of proposed methods. [ABSTRACT FROM AUTHOR]

In this paper, we present the general Bernstein functions for the first time. The properties of generalized Bernstein basis functions are given and demonstrated. The classical Bernstein polynomial bases are merely a subset of the general Bernstein functions. Based on the new Bernstein base functions and the collocation method, we present a numerical method for solving linear and nonlinear χ-fractional differential equations (χ-FDEs) with variable coefficients. The fractional derivative used in this work is the χ-Caputo fractional derivative sense (χ-CFD). Combining the Bernstein functions basis and the collocation methods yields the approximation solution of nonlinear differential equations. These base functions can be used to solve many problems in applied mathematics, including calculus of variations, differential equations, optimal control, and integral equations. Furthermore, the convergence of the method is rigorously justified and supported by numerical experiments. [ABSTRACT FROM AUTHOR]

The nonlinear Schrödinger equation (NLSE) is a fundamental equation in the field of nonlinear optics and plays an important role in the study of many physical phenomena. The present study introduces a new model that demonstrates the novelty of the paper and provides the advancement of knowledge in the area of nonlinear optics by solving a challenging problem known as the extended (3 + 1)-dimensional nonlinear conformable Kudryashov's equation (CKE) with generalized anti-cubic nonlinearity, which is a generalization of the NLSE to three spatial dimension and one temporal dimension for the first time. This work is significant because it advances our understanding of nonlinear optics and its applications to solve complex equations in physics and related disciplines. The extended hyperbolic function method (EHFM) and Nucci's reduction method are applied to the extended (3 + 1)-dimensional nonlinear CKE with generalized anti-cubic nonlinearity. The equation is solved by using the concept of conformable derivative, a recently developed operator in fractional calculus, which has advantages over other fractional derivatives in terms of accuracy and flexibility. The attained solutions include periodic singular, dark 1-soliton, singular 1-soliton, and bright 1-soliton which are visualized using 3D and contour plots. This study highlights the potential of using conformable derivative and the applied techniques to solve complex nonlinear differential equations in various fields. The obtained solutions and analysis will be useful in the design and analysis of optical communication systems and other related fields. Overall, this study contributes for the understanding of the dynamics of the extended (3+1)-dimensional nonlinear CKE and offers new insights into the use of mathematical techniques to tackle complex problems in physics and related fields. [ABSTRACT FROM AUTHOR]

This work develops a method for solving nonlinear evolution equations. The method, termed a bivariate spectral linear partition method (BSLPM), combines the Chebyshev spectral collocation method, bivariate Lagrange interpolation, and a linear partition technique as an underlying linearization method. It is developed for an nth-order nonlinear differential equation and then used to solve three known evolution problems. The results are compared with known exact solutions from literature. The method's applicability, reliability, and accuracy are confirmed by the congruence between the numerical and exact solutions. Tables, error graphs, and convergence graphs were generated using MATLAB (R2015a), to confirm the order of accuracy of the method and verify its convergence. The performance of the method is also observed against other methods performing well in these types of differential equations and is found to be comparable in terms of accuracy. The proposed method is also efficient as it uses minimal computation time. [ABSTRACT FROM AUTHOR]

This paper presents a new kinetic model with time-dependent proliferative/destructive parameters, where the activity variable of the system attains its values in a discrete real subset. Therefore, a system of nonautonomous nonlinear ordinary differential equations is gained, with the related Cauchy problem. A first result of local existence and uniqueness of positive and bounded solution is proved. Then, the possibility of extend this result globally in time is discussed, with respect to the shape of nonconservative time-dependent parameters. Numerical simulations are performed for some scenarios, corresponding to different shapes of time-dependent parameters themselves. Furthermore, the pattern and long-time behavior of solutions are numerically analyzed. Finally, these results show that equilibria and oscillations may occur. [ABSTRACT FROM AUTHOR]

In this work, we extend previous research on exponential integrators for stiff semilinear delay differential equations to the nonlinear case. In addition to the stiffness, there are two new issues that should be handled properly: nonlinear term and delay term. For nonlinear problems, a badly chosen linearization can cause a severe step size restriction. In this work, we linearize the equation along the numerical solution in each step. For the delay term, the interpolation based on the numerical values at the mesh points rather than inner stage values is adopted to significantly reduce the number of stiff order conditions. We focus on the construction and convergence analysis of high‐order exponential Rosenbrock methods for nonlinear stiff delay differential equations. The main result of this paper is that under the framework of strongly continuous semigroup, the explicit exponential Rosenbrock method is proved to be stiffly convergent of order p$$ p $$ even if the order conditions of order p$$ p $$ hold in a weak form. Moreover, by pointing that there does not exist fifth‐order method with less than or equal to four stages, we present the construction of a fifth‐order method with five stages. Finally, numerical tests are carried out to validate the theoretical results and to demonstrate the superiority of high‐order methods. [ABSTRACT FROM AUTHOR]

We propose an Adomian decomposition method to solve a class of nonlinear differential equations of fractional‐order with modified Caputo derivatives and integral boundary conditions. Our approach uses the integral boundary conditions to derive an equivalent nonlinear Volterra integral equation before establishing existence and uniqueness of solution and a recursion scheme for the solution. The convergence of the method is proved and an error analysis given. Two numerical examples are solved by obtaining a rapidly converging sequence of analytical functions to the solution. [ABSTRACT FROM AUTHOR]

Coronavirus has paralyzed various socio‐economic sectors worldwide. Such unprecedented outbreak was proved to be lethal for about 1,069,513 individuals based upon information released by Worldometers on October 09, 2020. In order to fathom transmission dynamics of the virus, different kinds of mathematical models have recently been proposed in literature. In the continuation, we have formulated a deterministic COVID‐19 model under fractional operators using six nonlinear ordinary differential equations. Using fixed‐point theory and Arzelá Ascoli principle, the proposed model is shown to have existence of unique solution while stability analysis for differential equations involved in the model is carried out via Ulam–Hyers and generalized Ulam–Hyers conditions in a Banach space. Real COVID‐19 cases considered from July 01 to August 14, 2020, in Pakistan were used to validate the model, thereby producing best fitted values for the parameters via nonlinear least‐squares approach while minimizing sum of squared residuals. Elasticity indices for each parameter are computed. Two numerical schemes under singular and non‐singular operators are formulated for the proposed model to obtain various simulations of particularly asymptomatically infectious individuals and of control reproduction number Rc. It has been shown that the fractional operators with order α=9.8254e−01 generated Rc=2.5087 which is smaller than the one obtained under the classical case (α=1). Interesting behavior of the virus is explained under fractional case for the epidemiologically relevant parameters. All results are illustrated from biological viewpoint. [ABSTRACT FROM AUTHOR]

In this study, we investigate and analyze an approximation of the Chebyshev polynomials for pantograph‐type fractional‐order differential equations. First, we construct the operational matrices of pantograph and Caputo fractional derivatives using Chebyshev interpolation. Then, the obtained matrices are utilized to approximate the fractional derivative. We also provide a detailed convergence analysis in terms of the weighted square norm. Finally, we describe and discuss the results of three numerical experiments conducted to confirm the applicability and accuracy of the computational scheme. [ABSTRACT FROM AUTHOR]

In this study, we present Taylor series solutions for steady‐state non‐isothermal diffusion–reaction problems pertaining to porous catalyst pellets exhibiting arbitrary kinetics. Using the Damkohler relation, the system of two nonlinear differential equations is reduced to a single differential equation subject to the algebraic constraint. We derive the novel semi‐analytical and closed‐form explicit approximate solutions for the reactant concentration and temperature in catalyst pellets of planar, cylindrical, and spherical geometries. The derived semi‐analytical and explicit approximations give insight into the effect of process parameters on concentration and temperature profiles. The proposed methods are verified numerically for isothermal and non‐isothermal steady‐state problems with power‐law kinetics. They can serve as a practicable alternative to numerical schemes. [ABSTRACT FROM AUTHOR]

Variable‐order fractional calculus operators can be used as a valuable tool to simulate many nonlinear models with a memory property in fractional calculus applications. In this paper, we developed a novel predictor–corrector method for solving numerically nonlinear differential equations with variable‐order Liouville–Caputo‐type fractional derivatives. First, we found an inversion integral formula that is equivalent to the studied problem, and then we used it to formulate our numerical algorithm. Numerical approximate solutions of some variable‐order fractional models have been presented to display the efficiency and accuracy of the proposed algorithm. The influence of the variable‐order on the dynamic behavior of the considered problems is described. [ABSTRACT FROM AUTHOR]

Conditions for the existence of a symmetric solution of nonlinear functional differential equations with symmetries are established. [ABSTRACT FROM AUTHOR]

In this paper, we devise a novel method to solve Kawahara‐type equations numerically. In this novel method, for spatial discretization, we use delta‐shaped basis functions and generate differentiation matrices for spatial derivatives of the Kawahara‐type equations. For discretization of temporal variable, we utilize a high‐order geometric numerical integrator based on Lie group methods. For illustration of efficiency of the suggested method, we consider some test problems. We calculate errors and make some comparisons with other results that exist in literature. We also report changes in conservation laws during numerical simulations, and we indicate that the suggested method can preserve the conservation laws pretty good. Outcomes of numerical simulations indicate that the suggested method in this paper is reliable and effective for nonlinear partial differential equations (PDEs). [ABSTRACT FROM AUTHOR]

Numerous classical models of thermostats have been analyzed in the literature, and while existence and uniqueness results have been established, the investigation of other properties, such as stability, can present significant challenges. Therefore, there is a need for fractional mathematical models that are nonlocal problems and allow for the straightforward study of the asymptotic behavior of their solutions. However, solving these problems is often exceedingly complex or even impossible, necessitating the use of numerical methods to obtain approximate solutions. The focus of this research is a generalized boundary value problem that is based on a classical thermostat model. Our initial step involves determining the associated integral equation for the problem at hand. The properties of existence and uniqueness are proved by exploiting the classical contraction principle of Banach together with another kind called ν$$ \nu $$‐ Φ$$ \Phi $$‐contraction. Afterwards, thanks to the Bernstein polynomials and fractional matrix of derivative, the approximate solutions of our nonlinear fractional thermostat model problem are obtained. This approach utilizes the Legendre operational matrix of fractional derivatives in combination with a truncated Legendre series for the numerical integration of fractional differential equations. By doing so, the problem can be reduced to solving a system of algebraic equations, resulting in significant simplification. This method has been successfully used to solve both linear and nonlinear fractional differential equations. Finally, an example on this type of problem have been studied and the approximate solution are plotted. [ABSTRACT FROM AUTHOR]

In this work, symmetry groups are used to determine symmetry reductions of a (2 + 1)‐dimensional Keller–Segel system depending on two arbitrary functions. We show that the point symmetries of the considered Keller–Segel system comprise an infinite‐dimensional Lie algebra which involves three arbitrary functions. By way of example, we have used these point symmetries to reduce straightaway the given system of second‐order partial differential equations to a system of second‐order ordinary differential equations. Moreover, we are allowed to substitute one of the dependent variables from one of the equations into the other, leading to an equivalent fourth‐order nonlinear ordinary differential equation. This equation is reduced through the use of solvable symmetry subalgebras, and some exact solutions are obtained for a particular case. [ABSTRACT FROM AUTHOR]

Nonlinear integro‐differential inequalities involving two arbitrary kernels are considered. Our results generalize some previous cases. The result about non‐existence of global solution in time has been proved. Our method of proof is based on the weak formulation of the problem. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present a new and efficient computational method based on the hybrid of block‐pulse functions and shifted Legendre polynomials together with their exact operational vector of integration and stochastic operational matrix of integration with respect to the multifractional Brownian motion to approximate solutions of a class of nonlinear stochastic differential equations driven by multifractional Gaussian noise. The presented method transforms problems under consideration into systems of nonlinear equations which can be simply solved by the Newton method. Moreover, the convergence analysis of the presented method is investigated. Finally, the efficiency of the new method is illustrated by solving the stochastic logistic equation and three test problems. [ABSTRACT FROM AUTHOR]

This article aims to investigate the electro‐osmotic flow of the Eyring‐Powell nanofluid under the peristaltic mechanism with the influence of the porous medium in the micro‐channel. The modified system is applied externally to an electrical field in the horizontal direction and a magnetic field in the transverse direction. Governing equations of nanofluid flow are formulated, and the assumptions of low Reynolds number and large wavelength approximations are tackled. The resulting coupled non‐linear partial differential equations like velocity and temperature equations are solved numerically by the NDSolve command using the computational mathematical software Mathematica. The influence of various important parameters on the velocity of the fluid, the temperature profiles, and the trapping phenomenon are discussed in detail by graphs. Under the influence of Joule heating, the study details the thermal transport process in the fluid. It demonstrates how an increase may significantly improve the momentum transmission in the core region of the microchannel in the volume fraction of the nanoparticles in the fluid. [ABSTRACT FROM AUTHOR]

This paper is concerned with the well‐posedness of a diffusion–reaction system for a susceptible‐exposed‐infected‐recovered (SEIR) mathematical model. This model is written in terms of four nonlinear partial differential equations with nonlinear diffusions, depending on the total amount of the SEIR populations. The model aims at describing the spatio‐temporal spread of the COVID‐19 pandemic and is a variation of the one recently introduced, discussed, and tested in a paper by Viguerie et al (2020). Here, we deal with the mathematical analysis of the resulting Cauchy–Neumann problem: The existence of solutions is proved in a rather general setting, and a suitable time discretization procedure is employed. It is worth mentioning that the uniform boundedness of the discrete solution is shown by carefully exploiting the structure of the system. Uniform estimates and passage to the limit with respect to the time step allow to complete the existence proof. Then, two uniqueness theorems are offered, one in the case of a constant diffusion coefficient and the other for more regular data, in combination with a regularity result for the solutions. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence and the uniqueness of solutions for a class of nonlinear fractional differential equations with a mixed fractional boundary value, by using Banach fixed‐point theorem. Moreover, we compare obtained results with another two works considering similar problem. [ABSTRACT FROM AUTHOR]

Consequence of thermal and concentration convection on peristaltic pumping of hyperbolic tangent nanofluid in a non‐uniform channel and induced magnetic field is discussed in this article. The brief mathematical modeling, along with induced magnetic field, of hyperbolic tangent nanofluid is given. The governing equations are reduced to dimensionless form by using appropriate transformations. Exact solutions are calculated for temperature, nanoparticle volume fraction, and concentration. Numerical technique is manipulated to solve the highly non‐linear differential equations. The roll of different variables is graphically analyzed in terms of concentration, temperature, volume fraction of nanoparticles, axial induced magnetic field, magnetic force function, stream functions, pressure rise, and pressure gradient. [ABSTRACT FROM AUTHOR]

The Eyring‐Powell fluid flow between two micro‐parallel plates in the context of electro‐magneto‐hydrodynamic is the focus of the article. The Lorentz force, which is generated by the interactions of a vertical magnetic field and an externally imposed horizontal electrical field, is presumed to be unilateral and one‐dimensional. Using the Darcy‐Brinkman‐Forchheimer model, the medium of the micro‐parallel plates is assumed to be porous. The energy equation evaluates the effect of viscous dissipation and joule heating as well. To solve the nonlinear coupled differential equations, analytical solutions are derived using the multi‐step differential transform method. For the velocity and temperature profiles, the impact of all evolving variables is explored and illustrated in graphs and tables. It can be seen from the graphical results that the Darcy parameter, and also the magnetic field, are oppositional to both fluid motion and temperature profile. Additionally, the thermal Grashof number improves the velocity and temperature profiles of the system. On the velocity and temperature profiles, the outcomes for both fluid parameters are nearly identical. [ABSTRACT FROM AUTHOR]

The present study examines the impact of unsteady viscous flow in a squeezing channel. Silver–gold hybrid nanofluid particles with different shapes are inserted in the base fluid engine oil. Flow and heat transfer mechanism is detected in the presence of magnetohydrodynamics between the two parallel infinite plates. The thermal conductivity models, that is, Yamada–Ota and Hamilton–Crosser models are used to investigate various shapes (Blade, platelet, cylinder, and brick) of hybrid nanoparticles. The model is made up of paired high nonlinear partial differential equations that are then transformed into ordinary differential equations which are coupled and strong nonlinear using the boundary layer approximation. The MATLAB solver bvp4c package is used to solve the numerical solution of this coupled system. The influence of different parameters on the physical quantities is addressed via graphs. A comparison with already reported results is given in order to confirm the current findings. The analysis shows that surprisingly the Yamada–Ota model of the Hybrid nanofluid gains high temperature and velocity profile than the Hamilton–Crosser model of the hybrid nanofluid. Also, both the models show increasing trends toward increasing the volume fraction rate of silver‐gold hybrid nanoparticles. It is also inferred that the hybrid‐nanoparticles performance is far better than the common nanofluids. [ABSTRACT FROM AUTHOR]

Because of thermal characteristics and the intention of improving heat transfer rate, the issue of hybrid nanofluid has become an intriguing subject for academicians and researchers in the contemporary technological period. This report presents the impacts of thermal radiation and viscous dissipation on the motion of the hybrid nanofluid of (Cu + Al2O3)‐H2O over a moving surface. The present flow model is simulated by a set of nonlinear partial differential equations (PDEs), which are converted into a system of ordinary differential equations (ODEs) by the appropriate similarity transformations. After that, the obtained ODEs are resolved numerically by bvp4c in MATLAB. The thermal radiation parameter has been shown to improve the rate of heat transfer and increase the temperature of the hybrid nanofluid (Cu + Al2O3)‐H2O. Moreover, two branches of solutions that depend on the stretching/shrinking parameter values are noticed. Furthermore, the stability results show that the upper branch can be used in practical applications, due to its stable nature. [ABSTRACT FROM AUTHOR]

This paper investigates a new class of p‐Laplacian boundary value problems of Hadamard type fractional differential equations. We emphasize that few works on Hadamard fractional boundary value problems include p‐Laplacian Hadamard type fractional differential equations with nonlinear terms involving Hadamard fractional derivatives of unknown functions. By means of the five functional fixed point theorem, multiple positive solutions are established for the p‐Laplacian Hadamard fractional boundary value problem. An example is constructed for illustration of the main result. [ABSTRACT FROM AUTHOR]

We perform the mathematical analysis of a model describing the interaction of two species in a chemostat, involving competition and mutualism, simultaneously. The model is a five‐dimensional system of differential equations with nonlinear growth functions. We give a comprehensive description of the dynamics of the system by determining analytically the existence and the local stability conditions of all steady‐states, considering a large class of growth rates. We prove that there exists a unique stable coexistence steady‐state and give the conditions under which bistability can occur. We give bifurcation diagrams and operating diagrams showing the rich behavior of the system. [ABSTRACT FROM AUTHOR]

In this article, we study existence, uniqueness, and stability analysis in Hyer–Ulam settings of delay fractional differential equations with nonlinear singular p‐Laplacian operator ϕp. The fractional operators are taken in the Caputo sense. The fractional differential equation is converted into an integral form with the Green's theorem, also a fixed point approach is studied for this article. We include an example with specific parameter and assumptions to show the results of the proposal. Our results generalize some of the works in the literature. [ABSTRACT FROM AUTHOR]

This paper deals with a nonlinear hybrid differential equation written using a fractional derivative with a Mittag–Leffler kernel. Firstly, we establish the existence of solutions to the studied problem by using the Banach contraction theorem. Then, by means of the Dhage fixed‐point principle, we discuss the existence of mild solutions. Finally, we study the Ulam–Hyers stability of the introduced fractional hybrid problem. [ABSTRACT FROM AUTHOR]

The well‐known Drinfeld–Sokolov–Wilson system is widely used to describe the flow of the shallow water. This paper considers the time–space fractional Drinfeld–Sokolov–Wilson system, where He's variational method together with the two‐scale transform is applied to find its solitary solutions. The numerical results have been shown through graphs to demonstrate the applicability and efficiency of our approach. The main advantage of variational approach is that it can reduce the order of differential equation and make the equation more simple. The obtained results in this work are expected to shed a bright light on the study of fractional nonlinear partial differential equations. [ABSTRACT FROM AUTHOR]

The first symptomatic infected individuals of coronavirus (Covid‐19) was confirmed in December 2020 in the city of Wuhan, China. In India, the first reported case of Covid‐19 was confirmed on 30 January 2020. Today, coronavirus has been spread out all over the world. In this manuscript, we studied the coronavirus epidemic model with a true data of India by using Predictor‐Corrector scheme. For the proposed model of Covid‐19, the numerical and graphical simulations are performed in a framework of the new generalised Caputo sense non‐integer order derivative. We analysed the existence and uniqueness of solution of the given fractional model by the definition of Chebyshev norm, Banach space, Schauder's second fixed point theorem, Arzel's‐Ascoli theorem, uniform boundedness, equicontinuity and Weissinger's fixed point theorem. A new analysis of the given model with the true data is given to analyse the dynamics of the model in fractional sense. Graphical simulations show the structure of the given classes of the non‐linear model with respect to the time variable. We investigated that the mentioned method is copiously strong and smooth to implement on the systems of non‐linear fractional differential equation systems. The stability results for the projected algorithm is also performed with the applications of some important lemmas. The present study gives the applicability of this new generalised version of Caputo type non‐integer operator in mathematical epidemiology. We compared that the fractional order results are more credible to the integer order results. [ABSTRACT FROM AUTHOR]

In this paper, we study a general nonautonomous model for bacterial dynamics in rivers. The mathematical model is represented by a nonautonomous system of nonlinear ordinary differential equations. We show the existence of a bounded positive invariant and attracting set. By using the Lyapunov function method, we establish global stability of steady‐state solutions of the associated autonomous system. Second, the existence of positive periodic solutions of the nonautonomous system is proven using a continuation theorem based on coincidence degree theory. [ABSTRACT FROM AUTHOR]

We present Wong‐type oscillation criteria for nonlinear impulsive differential equations having discontinuous solutions and involving both negative and positive coefficients. We use a technique that involves the use of a nonprincipal solution of the associated linear homogeneous equation. The existence of principal and nonpricipal solutions was recently obtained by the present authors. As in special cases, we have superlinear and sublinear Emden–Fowler equations under impulse effects. It is shown that the oscillatory behavior may change due to impulses. An example is also given to illustrate the importance of the results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the three‐dimensional non‐linear time‐fractional generalized Zakharov‐Kuznetsov (Z‐K) equation that describes the dust‐ion acoustic solitary waves in a magnetized dusty plasma. Based on the definition of the Riemann‐Liouville (R‐L) fractional derivatives, the analysis of the Lie group method can be applied to find the symmetries for the considered equation. After that, these symmetries enable us to transform the equation under study into a two‐dimensional non‐linear ordinary differential equation of fractional order in the sense of ErdLélyi‐Kober (E‐K) fractional operator. Also, we obtain a set of new analytical solutions for the time‐fractional generalized Z‐K equation via the power series and the fractional sub‐equation methods. Furthermore, we compute the conservation laws for this equation with a detailed derivation based on the formal Lagrangian. [ABSTRACT FROM AUTHOR]

We investigate the repeated averaging model for money exchanges: two agents picked uniformly at random share half of their wealth to each other. It is intuitively convincing that a Dirac distribution of wealth (centered at the initial average wealth) will be the long time equilibrium for this dynamics. In other words, the Gini index should converge to zero. To better understand this dynamics, we investigate its limit as the number of agents goes to infinity by proving the so‐called propagation of chaos, which links the stochastic agent‐based dynamics to a (limiting) nonlinear partial differential equation (PDE). This deterministic description has a flavor of the classical Boltzmann equation arising from statistical mechanics of dilute gases. We prove its convergence towards its Dirac equilibrium distribution by showing that the associated Gini index of the wealth distribution converges to zero with an explicit rate. [ABSTRACT FROM AUTHOR]

The generalized Korteweg–de Vries–Burgers–Kuramoto–Sivashinsky equation (GKdVBKS) is a nonlinear partial differential equation that models propagation of waves in a thick elastic tube filled with a viscous fluid. First, the dynamics of the GKdVBKS equation that depend on the physical parameters ν$$ \nu $$ and γ2$$ {\gamma}_2 $$, where γ2$$ {\gamma}_2 $$ is a positive parameter and ν$$ \nu $$ is a real number, are presented. We show that the set of constant equilibrium solutions is unstable when ν<−γ2$$ \nu <-{\gamma}_2 $$, Lyapunov stable when ν=−γ2$$ \nu =-{\gamma}_2 $$, and globally exponentially stable when ν>−γ2$$ \nu >-{\gamma}_2 $$. Second, we show that the controlled GKdVBKS equation has a unique strong solution. Then, we synthesize a single bounded input‐feedback controller for the GKdVBKS equation through its linearized system avoiding any spillover. The approach used is valid due to the Fréchet differentiability of the nonlinear C0$$ {C}_0 $$‐semigroup generated by the nonlinear GKdVBKS equation with its derivative being the linear C0$$ {C}_0 $$‐semigroup generated by the linearized system. Finally, numerical results for different values of the parameters of the equation are presented to illustrate the developed theory. [ABSTRACT FROM AUTHOR]

In the paper, we apply the generalized polynomial chaos expansion and spectral methods to the Burgers equation with a random perturbation on its left boundary condition. Firstly, the stochastic Galerkin method combined with the Legendre–Galerkin Chebyshev collocation scheme is adopted, which means that the original equation is transformed to the deterministic nonlinear equations by the stochastic Galerkin method and the Legendre–Galerkin Chebyshev collocation scheme is used to deal with the resulting nonlinear equations. Secondly, the stochastic Legendre–Galerkin Chebyshev collocation scheme is developed for solving the stochastic Burgers equation; that is, the stochastic Legendre–Galerkin method is used to discrete the random variable meanwhile the nonlinear term is interpolated through the Chebyshev–Gauss points. Then a set of deterministic linear equations can be obtained, which is in contrast to the other existing methods for the stochastic Burgers equation. The mean square convergence of the former method is analyzed. Numerical experiments are performed to show the effectiveness of our two methods. Both methods provide alternative approaches to deal with the stochastic differential equations with nonlinear terms. [ABSTRACT FROM AUTHOR]