Power series solution of linear and nonlinear partial differential equations via symbolic programming

In this paper, we introduce a series solution to a class of hyperbolic system of time-fractional partial differential equations with variable coefficients. The fractional derivative has been considered by the concept of Caputo. Two expansions of matrix functions are proposed and used to create series solutions for the target problem. The first one is a fractional Laurent series, and the second is a fractional power series. A new approach, via the residual power series method and the Laplace transform, is also used to find the coefficients of the series solution. In order to test our proposed method, we discuss four interesting and important applications. Numerical results are given to authenticate the efficiency and accuracy of our method and to test the validity of our obtained results. Moreover, solution surface graphs are plotted to illustrate the effect of fractional derivative arrangement on the behavior of the solution.

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Analytical Solutions for the Nonlinear Partial Differential Equations Using the Conformable Triple Laplace Transform Decomposition Method

International Journal of Differential Equations ◽

10.1155/2021/9988160 ◽

2021 ◽

Vol 2021 ◽

pp. 1-18

Author(s):

Shailesh A. Bhanotar ◽

Mohammed K. A. Kaabar

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Laplace Transform ◽

Analytical Solutions ◽

Decomposition Method ◽

Nonlinear Partial Differential Equations ◽

Three Dimensions ◽

Conformable Derivative ◽

Partial Differential ◽

Linear And Nonlinear

In this paper, a novel analytical method for solving nonlinear partial differential equations is studied. This method is known as triple Laplace transform decomposition method. This method is generalized in the sense of conformable derivative. Important results and theorems concerning this method are discussed. A new algorithm is proposed to solve linear and nonlinear partial differential equations in three dimensions. Moreover, some examples are provided to verify the performance of the proposed algorithm. This method presents a wide applicability to solve nonlinear partial differential equations in the sense of conformable derivative.

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Continuous Solution of Partial Differential Equations on Non-Rectangular and Non-Continuous Geometry

Volume 1A: 34th Computers and Information in Engineering Conference ◽

10.1115/detc2014-35148 ◽

2014 ◽

Author(s):

P. Venkataraman

Keyword(s):

Finite Element ◽

Partial Differential Equations ◽

Differential Equations ◽

Continuous Solution ◽

Finite Difference Methods ◽

Nonlinear Partial Differential Equations ◽

Partial Differential ◽

Mesh Free ◽

Numeric Calculation ◽

Linear And Nonlinear

A high order continuous solution is obtained for partial differential equations on non-rectangular and non-continuous domain using Bézier functions. This is a mesh free alternative to finite element or finite difference methods that are normally used to solve such problems. The problem is handled without any transformation and the setup is direct, simple, and involves minimizing the error in the residuals of the differential equations along with the error in the boundary conditions over the domain. The solution can be expressed in polynomial form. The effort is same for linear and nonlinear partial differential equations. The procedure is developed as a combination of symbolic and numeric calculation. The solution is obtained through the application of standard unconstrained optimization. A constrained approach is also developed for nonlinear partial differential equations. Examples include linear and nonlinear partial differential equations. The solution for linear partial differential equations is compared to finite element solutions from COMSOL.

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Techniques in Linear and Nonlinear Partial Differential Equations.

10.21236/ada187109 ◽

1987 ◽

Author(s):

Louis Nirenberg

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Nonlinear Partial Differential Equations ◽

Partial Differential ◽

Linear And Nonlinear

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Power Series Solution for Solving Nonlinear Burgers-Type Equations

Abstract and Applied Analysis ◽

10.1155/2015/712584 ◽

2015 ◽

Vol 2015 ◽

pp. 1-9 ◽

Cited By ~ 1

Author(s):

E. López-Sandoval ◽

A. Mello ◽

J. J. Godina-Nava ◽

A. R. Samana

Keyword(s):

Power Series ◽

Differential Equations ◽

Series Solution ◽

Solution Method ◽

Nonlinear Partial Differential Equations ◽

Time Dependent ◽

Power Series Solution ◽

Partial Linear ◽

Linear Differential ◽

Burgers Type Equations

Power series solution method has been traditionally used to solve ordinary and partial linear differential equations. However, despite their usefulness the application of this method has been limited to this particular kind of equations. In this work we use the method of power series to solve nonlinear partial differential equations. The method is applied to solve three versions of nonlinear time-dependent Burgers-type differential equations in order to demonstrate its scope and applicability.

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