Method For Nonlinear Differential Equations Research Articles

The traveling wave solutions to a variant of the Boussinesq equation

In this study, we study a variant of the Boussinesq equation called as \(B( n+1,\,1,\,n )\) equation, and construct some traveling wave solutions by using an effective approach called the extended trial equation method. Thus, the soliton solutions, rational function solutions, elliptic function solutions and Jacobi elliptic function solutions, which show the existence of various mathematical and physical structures and events in the fundamental equation considered, have been constructured. In order to make a more detailed examination of the physical behavior of these solutions, two- and three-dimensional graphs of some solution functions were drawn with the help of the Mathematica package program. In the section of Discussion, we suggest a more general version of the trial equation method for nonlinear differential equations.

A successive midpoint method for nonlinear differential equations with classical and Caputo-Fabrizio derivatives

<abstract><p>In this study, we present a numerical scheme for solving nonlinear ordinary differential equations with classical and Caputo–Fabrizio derivatives using consecutive interval division and the midpoint approach. By doing so, we increased the accuracy of the midpoint approach, which is dependent on the number of interval divisions. In the example of the Caputo–Fabrizio differential operator, we established the existence and uniqueness of the solution using the Caratheodory-Tonelli sequence. We solved numerous nonlinear equations and determined the global error to test the accuracy of the proposed scheme. When the differential equation met the circumstances under which it was generated, the results revealed that the procedure was quite accurate.</p> </abstract>

Aboodh transform homotopy perturbation method for solving fractional‐order Newell‐Whitehead‐Segel equation

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.

Study on approximate analytical methods for nonlinear differential equations

In this work, an analytical approximation solution is presented, as well as a comparison of the Variational Iteration Adomian Decomposition Method (VIADM) and the Modified Sumudu Transform Adomian Decomposition Method (M STADM), both of which are capable of solving nonlinear partial differential equations (NPDEs) such as nonhomogeneous Kertewege-de Vries (kdv) problems and the nonlinear Klein-Gordon. The results demonstrate the solution’s dependability and excellent accuracy.

Doubly stochastic Yule cascades (Part I): The explosion problem in the time-reversible case

Motivated by the probabilistic methods for nonlinear differential equations introduced by McKean (1975) for the Kolmogorov-Petrovski-Piskunov (KPP) equation, and by Le Jan and Sznitman (1997) for the incompressible Navier-Stokes equations (NSE), we identify a new class of stochastic cascade models, referred to as doubly stochastic Yule cascades. We establish non-explosion criteria under the assumption that the randomization of Yule intensities from generation to generation is by an ergodic time-reversible Markov process. In addition to the cascade models that arise in the analysis of certain deterministic nonlinear differential equations, this model includes the multiplicative branching random walks, the branching Markov processes, and the stochastic generalizations of the percolation and/or cell ageing models introduced by Aldous and Shields (1988) and independently by Athreya (1985).

A Haar wavelet multi-resolution collocation method for singularly perturbed differential equations with integral boundary conditions

The focus of this paper is to develop and improve a higher-order Haar wavelet approach for solving nonlinear singularly perturbed differential equations with various pairs of boundary conditions like initial, boundary, two points, integral and multi-point integral boundary conditions. The theoretical convergence and computational stability of the method is also presented. The comparison of the proposed higher-order Haar wavelet method is performed with the recent published work including the well-known Haar wavelet method in terms of convergence and accuracy. In the nonlinear case, a quasilinearization technique has been adopted. The proposed method is easy to implement on various boundary conditions, and the computed results are high-order accurate, stable and efficient. We have also checked the satisfactory performance of the proposed method for nonlinear differential equations having no analytical solution in some of the test problems.

Numerical investigation of generalized perturbed Zakharov–Kuznetsov equation of fractional order in dusty plasma

In the present work, the new iterative method with a combination of the Laplace transform of the Caputo’s fractional derivative has been applied to the generalized (3 + 1) dimensional fractional perturbed Zakharov–Kuznetsov equation in a dusty plasma. The proposed method is applied without any discretization and linearization. The numerical and graphical results show the accuracy of the proposed method for nonlinear differential equations. Moreover, the methods are easy to implement and give the efficient approximate solutions.

A REVIEW ON NUMERICAL METHODS FOR NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS

Progression in innovation and engineering presents us with numerous difficulties, comparably to conquer such engineering difficulties with the assistance of various numerical models, equations are taken. Since in the first place Mathematicians, Designers and Engineers make progress toward accuracy what’s more, exactness while addressing equations Differential equations, specifically, hold an enormous application in engineering and numerous different areas. One such sort of Differential equation is known as partial differential equation. The range of application of partial differential equations comprises of recreation, calculation age, and investigation of higher request PDE and wave equations. Adjusting diverse numerical methods prompts an assortment of answers and contrast among them, subsequently the determination of the method of addressing is one of the urgent boundaries to produce exact outcomes. Our work centres’ around the survey of various numerical methods to settle Non-linear differential equations based on exactness and effectiveness, in order to diminish the emphases. These would orchestrate rules to existing numerical methods of nonlinear partial differential equations.[1]

Study of Convergence of Reduced Differential Transform Method for Different Classes of Differential Equations

In this work, we study the sufficient condition for convergence of the reduced differential transform method for nonlinear differential equations. The main power of this method is its ability and flexibility in solving linear and nonlinear problems properly and easily and obtain solutions both numerically and analytically. Simple approaches of reduced differential transform method and the convergence results for different classes of differential equations such as linear and nonlinear ordinary, partial, fractional, and system of differential equations are briefly discussed. Eight examples are checked to confirm convergence results as well as the strength and efficiency of the method.

A B-spline finite element method for nonlinear differential equations describing crystal surface growth with variable coefficient

In this paper, an efficient finite element scheme is presented for a class of fourth-order nonlinear parabolic problems with variable coefficient. To deal with second-order term in weak formulation, we choose the cubic B-spline function as a trial function. Rigorous error estimates are derived for both semi-discrete and fully-discrete schemes. We provide a numerical example to confirm our theoretical results.

Dual solutions for three-dimensional magnetohydrodynamic nanofluid flow with entropy generation

Abstract This paper presents a formulation for simulating magnetohydrodynamic three-dimensional convective flow and heat transfer in a nanofluid by incorporating the complete viscous dissipation function in the energy equation. A novel feature of this investigation of entropy generation and dual solutions is the use of the spectral quasilinearization method to solve the conservation equations. The results are compared with exact solutions or higher order solutions and a good agreement is achieved. The accuracy is determined by calculation of residual errors and the method of solution is shown to produce smaller residual errors than those achieved by the fifth-order Runge-Kutta Fehlberg method for nonlinear differential equations. The dual solutions for different Prandtl number, and Brownian motion and thermophoresis parameters are shown graphically and discussed. It is found that the temperature profiles as well as thermal boundary layer thickness increase with the Brownian motion parameter for first and the second solutions. The temperature profiles increase with the thermophoresis parameter for the first and second solutions. The entropy generation increases with the Reynolds number. Highlights Combined effects of entropy generation and MHD nanofluid are proposed. Spectral quasi-linearization method (SQLM) is used for computer simulations. Use axisymmetric stretching/shrinking sheet for dual solution. Validate the accuracy and convergence using residual error analysis.

First integral method for non-linear differential equations with conformable derivative

In this paper, we present an analysis based on the first integral method in order to construct exact solutions of the nonlinear fractional partial differential equations (FPDE) described by beta-derivative. A general scheme to find the approximated solutions of the nonlinear FPDE is showed. The results obtained showed that the first integral method is an efficient technique for analytic treatment of nonlinear beta-derivative FPDE.

Optimal homotopy perturbation method for nonlinear differential equations governing MHD Jeffery-Hamel flow with heat transfer problem

Abstract In this paper, the Optimal Homotopy Perturbation Method (OHPM) is employed to determine an analytic approximate solution for the nonlinear MHD Jeffery-Hamel flow and heat transfer problem. The Navier-Stokes equations, taking into account Maxwell’s electromagnetism and heat transfer, lead to two nonlinear ordinary differential equations. The results obtained by means of OHPM show very good agreement with numerical results and with Homotopy Perturbation Method (HPM) results.

Homotopy perturbation transform method for nonlinear differential equations involving to fractional operator with exponential kernel

This work presents the homotopy perturbation transform method for nonlinear fractional partial differential equations of the Caputo-Fabrizio fractional operator. Perturbative expansion polynomials are considered to obtain an infinite series solution. The effectiveness of this method is demonstrated by finding the exact solutions of the fractional equations proposed, for the special case when the limit of the integral order of the time derivative is considered.

Quasilinearization method for nonlinear differential equations with causal operators

Quasilinearization method for nonlinear differential equations with causal operators

INVESTIGATION OF OSCILLATORY SOLUTIONS OF DIFFERENTIAL-DIFFERENCE EQUATIONS OF SECOND ORDER IN A CRITICAL CASE

We consider a differential-difference equation of second order of delay type, containing the delay of the function and its derivatives. Such equations occur in the modeling of electronic devices. The nature of the loss of the zero solution stability is studied. The possibility of stability loss related to the passing of two pairs of purely imaginary roots, that are in resonance 1:3, through an imaginary axis is shown. In this case bifurcating oscillatory solutions are studied. It is noted the existence of a chaotic attractor for which Lyapunov exponents and Lyapunov dimension are calculated. As an investigation techniques we use the theory of integral manifolds and normal forms method for nonlinear differential equations.

On the analytical solution of Fornberg–Whitham equation with the new fractional derivative

Motivated by the simplicity, natural and efficient nature of the new fractional derivative introduced by R Khalil et al in J. Comput. Appl. Math. 264, 65 (2014), analytical solution of space-time fractional Fornberg–Whitham equation is obtained in series form using the relatively new method called q-homotopy analysis method (q-HAM). The new fractional derivative makes it possible to introduce fractional order in space to the Fornberg–Whitham equation and be able to obtain its solution. This work displays the elegant nature of the application of q-HAM to solve strongly nonlinear fractional differential equations. The presence of the auxiliary parameter h helps in an effective way to obtain better approximation comparable to exact solutions. The fraction-factor in this method gives it an edge over other existing analytical methods for nonlinear differential equations. Comparisons are made on the existence of exact solutions to these models. The analysis shows that our analytical solutions converge very rapidly to the exact solutions.

ON THE SOLUTIONS OF NON-LINEAR TIME-FRACTIONAL GAS DYNAMIC EQUATIONS: AN ANALYTICAL APPROACH

We consider non-linear homogeneous and non-homogeneous gas dynamic equations of time-fractional type in this paper. The approximate solutions of these equations are calculated in the form of series obtained by q-Homotopy Analysis Method (q-HAM). Exact solution is obtained for time- fractional homogeneous case while for the case of time-fractional non-homo- geneous, exact solution is possible for special case. This is due to the ability to control the auxiliary parameter h and the fraction factor present in this method. The presence of fraction-factor in this method gives it an edge over other exist- ing analytical methods for non-linear differential equations. Comparisons are made with several other analytical methods.

Asymptotic stability of Runge–Kutta methods for nonlinear differential equations with piecewise continuous arguments

This paper deals with the asymptotic stability of numerical solutions for differential equations with piecewise continuous arguments (EPCAs). The necessary and sufficient condition is given for non-confluent Runge–Kutta methods to preserve the stability of nonlinear scalar EPCAs. As for systems, we prove that some algebraically stable methods can preserve the asymptotic stability.

Analytical solutions of time-fractional models for homogeneous Gardner equation and non-homogeneous differential equations

In this paper, we obtain analytical solutions of homogeneous time-fractional Gardner equation and non-homogeneous time-fractional models (including Buck-master equation) using q-Homotopy Analysis Method (q-HAM). Our work displays the elegant nature of the application of q-HAM not only to solve homogeneous non-linear fractional differential equations but also to solve the non-homogeneous fractional differential equations. The presence of the auxiliary parameter h helps in an effective way to obtain better approximation comparable to exact solutions. The fraction-factor in this method gives it an edge over other existing analytical methods for non-linear differential equations. Comparisons are made upon the existence of exact solutions to these models. The analysis shows that our analytical solutions converge very rapidly to the exact solutions.