The Combined Laplace-homotopy Analysis Method for Partial Differential Equations

In the paper, two new analytic methods using the decomposition and linearization technique on nonlinear differential/integral equations are developed, namely, the decomposition–linearization–sequential method (DLSM) and the linearized homotopy perturbation method (LHPM). The DLSM is realized by an integrating factor and the integral of certain function obtained at the previous step for obtaining a sequential analytic solution of nonlinear differential equation, which provides quite accurate analytic solution. Some first- and second-order nonlinear differential equations display the fast convergence and accuracy of the DLSM. An analytic approximation for the Volterra differential–integral equation model of the population growth of a species is obtained by using the LHPM. In addition, the LHPM is also applied to the first-, second-, and third-order nonlinear ordinary differential equations. To reduce the cost of computation of He’s homotopy perturbation method and enhance the accuracy for solving cubically nonlinear jerk equations, the LHPM is implemented by invoking a linearization technique in advance is developed. A generalization of the LHPM to the nth-order nonlinear differential equation is involved, which can greatly simplify the work to find an analytic solution by solving a set of second-order linear differential equations. A remarkable feature of those new analytic methods is that just a few steps and lower-order approximations are sufficient for producing reasonably accurate analytic solutions. For all examples, the second-order analytic solution x2(t) is found to be a good approximation of the real solution. The accuracy of the obtained approximate solutions are identified by the fourth-order Runge–Kutta method. The major objection is to unify the analytic solution methods of different nonlinear differential equations by simply solving a set of first-order or second-order linear differential equations. It is clear that the new technique considerably saves computational costs and converges faster than other analytical solution techniques existing in the literature, including the Picard iteration method. Moreover, the accuracy of the obtained analytic solution is raised.

A new treatment for homotopy perturbation method is introduced. The new treatment is called He‐Laplace method which is the coupling of the Laplace transform and the homotopy perturbation method using He’s polynomials. The nonlinear terms can be easily handled by the use of He’s polynomials. The method is implemented on linear and nonlinear partial differential equations. It is found that the proposed scheme provides the solution without any discretization or restrictive assumptions and avoids the round‐off errors.

In this paper, a novel technique, that is to read, the New Homotopy Perturbation Method (NHPM) is utilized for solving a linear and non-linear differential equations and integral equations. The two most important steps in the application of the new homotopy perturbation method are to invent a suitable homotopy equation and to choose a suitable initial conditions. Comparing between the effects of the method (NHPM), is given exact solution, and the method (HPM), is given approximate solution, in this paper, we make some instances are provided to prove the ability of the method (NHPM). Show that the method (NHPM) is valid and effective, easy and accurate in solving linear and nonlinear differential equations, compared with the Homotopy Perturbation Method (HPM).

In this paper, the weighted average-based differential quadrature method is presented for solving one-dimensional homogeneous first-order non-linear parabolic partial differential equation. First, the given solution domain is discretized by using uniform discretization grid point. Next, by using Taylor series expansion we obtain central finite difference discretization of the partial differential equation involving with temporal variable associated with weighted average of partial derivative concerning spatial variable. From this, we obtain the system of nonlinear ordinary differential equations and it is linearized by using the quasilinearization method. Then by using the polynomial-based differential quadrature method for approximating derivative involving with spatial variable at specified grid point, we obtain the system of linear equation. Then they obtained linear system equation is solved by using the LU matrix decomposition method. To validate the applicability of the proposed method, two model examples are considered and solved at each specific grid point on its solution domain. The stability and convergent analysis of the present method is worked by supported the theoretical and mathematical statements and the accuracy of the solution is obtained. The accuracy of the present method has been shown in the sense of root mean square error norm and maximum absolute error norm and the local behavior of the solution is captured exactly. Numerical versus exact solutions and behavior of maximum absolute error between them have been presented in terms of graphs and the corresponding root mean square error norm and maximum absolute error norm presented in tables. The present method approximates the exact solution very well and it is quite efficient and practically well suited for solving the non-linear parabolic equation. The numerical result presented in tables and graphs indicates that the approximate solution is in good agreement with the exact solution.

Homotopy perturbation method (HPM) is a semi-analytical technique for solving linear as well as nonlinear ordinary/partial differential equations. The method may also be used to solve a system of coupled linear and nonlinear differential equations. This chapter first illustrates the basic idea of the HPM. It uses the present method to solve a nonlinear ordinary differential equation, a nonlinear partial differential equation, and a coupled nonlinear system of partial differential equations. It is worth mentioning that the HPM can also be used for handling linear ordinary and partial differential equations. The test problems demonstrated in the chapter confirm that the HPM is an efficient method for solving linear/nonlinear ordinary, partial, and coupled differential equations. The series usually converges with an increase in the number of terms, but one may not expect the compact form solution always.

We apply the homotopy perturbation method to obtain the solution of partial differential equations of fractional order. This method is powerful tool to find exact and approximate solution of many linear and nonlinear partial differential equations of fractional order. Convergence of the method is proved and the convergence analysis is reliable enough to estimate the maximum absolute truncated error of the series solution. The fractional derivatives are described in the Caputo sense. Some examples are presented to verify convergence hypothesis and simplicity of the method.

Solution of nonlinear partial differential equations from base equations

PurposeThis paper aims to solve linear and non-linear shallow water wave equations using homotopy perturbation method (HPM). HPM is a straightforward method to handle linear and non-linear differential equations. As such here, one-dimensional shallow water wave equations have been considered to solve those by HPM. Interesting results are reported when the solutions of linear and non-linear equations are compared.Design/methodology/approachHPM was used in this study.FindingsSolution of one-dimensional linear and non-linear shallow water wave equations and comparison of linear and non-linear coupled shallow water waves from the results obtained using present method.Originality/valueCoupled non-linear shallow water wave equations are solved.

An elegant and powerful technique is Homotopy Perturbation Method (HPM) to solve linear and nonlinear ordinary and partial differential equations. The method, which is a coupling of the traditional perturbation method and homotopy in topology, deforms continuously to a simple problem which can be solved easily. The method does not depend upon a small parameter in the equation. Using the initial conditions this method provides an analytical or exact solution. From the calculation and its graphical representation it is clear that how the solution of the original equation and its behavior depends on the initial conditions. Therefore there have been attempts to develop new techniques for obtaining analytical solutions which reasonably approximate the exact solutions. Many problems in natural and engineering sciences are modeled by nonlinear partial differential equations (NPDEs). The theory of nonlinear problem has recently undergone much study. Nonlinear phenomena have important applications in applied mathematics, physics, and issues related to engineering. In this paper we have applied this method to Burger’s equation and an example of highly nonlinear partial differential equation to get the most accurate solutions. The final results tell us that the proposed method is more efficient and easier to handle when is compared with the exact solutions or Adomian Decomposition Method (ADM).

: Complex physical phenomena involving chemically reacting systems or the transport of heat or fluids are often modeled by coupled systems of time-dependent, nonlinear partial differential equations. The difficulties in understanding the stability of the differential equation systems and in designing efficient, accurate numerical methods for their solution are widely recognized and were the focus of this research. We have worked on four general aspects of the analysis and numerical approximation of systems of partial differential equations. These areas of research are: (1) modeling aspects and stability analysis for nonlinear time-dependent partial differential equations; (2) use and analysis of finite element or finite difference methods to discretize coupled systems of nonlinear differential equations; (3) development of adaptive or local grid refinement capabilities to resolve local phenomena in large-scale applications; and (4) development of data structures, preconditioners, and efficient solution algorithms for large-scale problems on new computer architectures. Emphasis has been placed upon multiphase or multicomponent, transport-dominated flow processes with dynamic local phenomena. The research also involved a mix of analysis, algorithm development, and large-scale computation using newer computer architectures.

ABSTRACTIn this article, we have established the homotopy analysis method (HAM) for solving a few partial differential equations arising in engineering. This technique provides the solutions in rapid convergence series with computable terms for the problems with high degree of nonlinear terms appearing in the governing differential equations. The convergence analysis of the proposed method is also discussed. Finally, we have given some illustrative examples to demonstrate the validity and applicability of the proposed method.

In this paper, the nonlinear free vibrations of thin symmetric and non-symmetric cross-ply composite plates subjected to biaxial initial stresses are investigated. Because of their excellent properties such as specific strength and specific stiffness, composite plates have wide applications in aerospace and mechanical structures. Based on Von-Karman's strain-displacement relations and using Galerkin method, the nonlinear differential equation of free vibrations of initially stressed composite plate is obtained. This nonlinear equation is solved using two different analytical perturbation methods, namely method of multiple scales (MTS) and homotopy perturbation method (HPM), to analyze the nonlinear vibrations of initially stressed cross-ply composite plates. Effects of tensile and compressive biaxial initial stresses, initial vibration amplitude, thickness, and aspect ratios of the composite plates on the frequency behavior are investigated. The validity of the results is confirmed by making a comparison with those reported in the literature. According to the results, both analytical solutions show increasing trends for natural frequency parameters by increasing normal initial stresses. Regardless of the value of initial biaxial stresses, for both symmetric and non-symmetric plates, the results of MTS and HPM are in close agreement for the smallest initial amplitude. However, for compressive initial stresses, by increasing initial amplitude ratios, the discrepancies between the results of HPM and MTS increase for symmetric and non-symmetric plates. Although HPM includes less computational effort (smaller length of formulation) than MTS, the linear-to-nonlinear frequency ratios obtained using MTS method become closer to those obtained by HPM as initial vibration amplitude is decreased and initial stress is increased.

In the present work, we apply the variational iteration method using He’s polynomials (VIMHP) for solving four examples of the biological population model (BPM). The proposed method is a combination of He’s variational iteration and the homotopy perturbation methods. The suggested algorithm is practically more reliable, highly efficient for use in such problems. The proposed method finds the solution without any restrictive assumptions, discretization and linearization. The approximate solution converges very rapidly to the exact solution which confirms the accuracy of this method as an easy algorithm for computing the solution for wider classes of linear and nonlinear differential equations.

The integral-differential equations describe many phenomena in different field of mechanical and nuclear engineering, chemistry, astronomy, biology and potential theory. The resolution of boundary problems for these equations are the subject of several papers in which the authors have approached in most cases numerical methods: Monte Carlo, truncated series of Chebyshev polynomials, the fictitious domain method, S N method and the finite element method (1-4). An exact solution was found only in the particular cases. Generally, these are obtained with the help of the methods of mathematical analysis, abstract functional analysis and the spectral methods (5). In recent years, the basic ideas of the homotopy, which is a concept of the topology and differential geometry, were used to obtain the approximate solutions for a wide class of differential, integral and integral-differential equations. We mention here the homotopy perturbation method (HPM) proposed by Ji-Huan He in 1998 and the homotopy analysis method (HAM) proposed by Liao in 1992. The perturbation methods approximate the solution of given problem by a series of small parameters. Unfortunately, the majority of non-linear problems have no small parameters and an unsuitable choice of these parameters can lead to bad effects. The new homotopy perturbation technique (HPM) embeds a parameter p that ranges from zero to one (6,7). When the embedding parameter is zero, we get a linear equation and if it is equal to one, we get the original transport equation. This embedding parameter that belongs to the interval (0, 1) can be considered as a small parameter. The homotopy analysis method (HAM) is one of the most effective methods to obtain the exact and approximate series solutions for the integral-differential equations (8). Being a great freedom in choosing of the initial guess functions and the auxiliary functions, are proposed for these such forms that correspond to the conditions imposed at the ends of the spatial interval and thus the algorithm becomes fast convergent. In the papers (9-11) is shown that the homotopy perturbation method, variational iteration method and Adomian decomposition method are only special cases of the homotopy analysis method solutions. Today, a new homotopy method is added to the above. With this we can solve the multipoint boundary problem for a stationary transport equation in the non-homogeneous media. Comparative studies between the solutions obtained with the help of mentioned algorithms for a numerical example with the appropriate exact solutions illustrate the accuracy and computational efficiency of these new methods.

In the present paper, we established a traveling wave solution by using modified Kudryashov method for the space-time fractional nonlinear partial differential equations. The method is used to obtain the exact solutions for different types of the space-time fractional nonlinear partial differential equations such as, the space-time fractional coupled equal width wave equation (CEWE) and the space-time fractional coupled modified equal width wave equation (CMEW), which are the important soliton equations. Both equations are reduced to ordinary differential equations by the use of fractional complex transform and properties of modified Riemann–Liouville derivative. We plot the exact solutions for these equations at different time levels.