scholarly journals Novel techniques for solving Goursat partial differential equations in the linear and nonlinear regime

2022 ◽  
Vol 1 (1) ◽  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Regime ◽  
Partial Differential ◽  
Linear And Nonlinear

scholarly journals Using Homo-Separation of Variables for Solving Systems of Nonlinear Fractional Partial Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Abdolamir Karbalaie ◽  
Hamed Hamid Muhammed ◽  
Bjorn-Erik Erlandsson
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Separation Of Variables ◽  
General Purpose ◽  
New Method ◽  
Fractional Partial Differential Equations ◽  
Mathematical Methods ◽  
Partial Differential ◽  
Separation Of Variables Method ◽  
Linear And Nonlinear

A new method proposed and coined by the authors as the homo-separation of variables method is utilized to solve systems of linear and nonlinear fractional partial differential equations (FPDEs). The new method is a combination of two well-established mathematical methods, namely, the homotopy perturbation method (HPM) and the separation of variables method. When compared to existing analytical and numerical methods, the method resulting from our approach shows that it is capable of simplifying the target problem at hand and reducing the computational load that is required to solve it, considerably. The efficiency and usefulness of this new general-purpose method is verified by several examples, where different systems of linear and nonlinear FPDEs are solved.

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

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.

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

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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