A general solution of some linear partial differential equations via two integral transforms (original) (raw)
A note on integral transforms and partial differential equations
2010
In this study, we apply double integral transforms to solve partial differential equation namely double Laplace and Sumudu transforms, in particular the wave and poisson's equations were solved by double Sumudu transform and the same result can be obtained by double Laplace transform.
2020
The primary purpose of this research is to demonstrate an efficient replacement double transform named the Laplace–Sumudu transform (DLST) to unravel integral differential equations. The theorems handling fashionable properties of the Laplace–Sumudu transform are proved; the convolution theorem with an evidence is mentioned; then, via the usage of these outcomes, the solution of integral differential equations is built.
International Journal of Analysis and Applications , 2019
In this paper, we introduce a Laplace-type integral transform called the Shehu transform which is a generalization of the Laplace and the Sumudu integral transforms for solving differential equations in the time domain. The proposed integral transform is successfully derived from the classical Fourier integral transform and is applied to both ordinary and partial differential equations to show its simplicity, efficiency, and the high accuracy.
Axioms
This article demonstrates how the new Double Laplace–Sumudu transform (DLST) is successfully implemented in combination with the iterative method to obtain the exact solutions of nonlinear partial differential equations (NLPDEs) by considering specified conditions. The solutions of nonlinear terms of these equations were determined by using the successive iterative procedure. The proposed technique has the advantage of generating exact solutions, and it is easy to apply analytically on the given problems. In addition, the theorems handling the mode properties of the DLST have been proved. To prove the usability and effectiveness of this method, examples have been given. The results show that the presented method holds promise for solving other types of NLPDEs.
On a new integral transform and differential equations
2010
Integral transform method is widely used to solve the several differential equations with the initial values or boundary conditions which are represented by integral equations. With this purpose, the Sumudu transform was introduced as a new integral transform by Watugala to solve some ordinary differential equations in control engineering. Later, it was proved that Sumudu transform has very special and useful properties. In this paper we study this interesting integral transform and its efficiency in solving the linear ordinary differential equations with constant and nonconstant coefficients as well as system of differential equations.
The combined Laplace transform-differential transform method for solving linear non-homogeneous PDEs
2012
In this work, a combined form of the Laplace transform method (LTM) with the differential transform method (DTM) will be used to solve non-homogeneous linear partial differential equations (PDEs). The combined method is capable of handling non-homogeneous linear partial differential equations with variable coefficient. The aim of using the Laplace transform is to overcome the deficiency that is caused by unsatisfied boundary conditions in using differential transform method. Illustrative examples will be examined to support the proposed analysis.
European Journal of Pure and Applied Mathematics
In this article, a new effective technique is implemented to solve families of nonlinear partial differential equations (NLPDEs). The proposed method combines the double ARA-Sumudu transform with the numerical iterative method to get the exact solutions of NLPDEs. The successive iterative method was used to find the solution of nonlinear terms of these equations. In order to show the efficiency and applicability of the presented method, some physical applications are analyzed and illustrated, and to defend our results, some numerical examples and figures are discussed.
Solutions of certain initial-boundary value problems via a new extended Laplace transform
Nonlinear engineering, 2024
In this article, we present a novel extended exponential kernel Laplace-type integral transform. The Laplace, natural, and Sumudu transforms are all included in the suggested transform. The existence theorem, Parseval-type identity, inversion formula, and other fundamental aspects of the new integral transform are examined in this article. Integral identities define the connections between the new transforms and the established transforms. In order to solve specific initial-boundary value problems, the new transforms are used.
IJRAR, 2019
In this article, first the properties of two dimensional differential transform method are presented. After this, by using the idea of two dimensional differential transform method we will find an analytical numerical solution of linear partial integro-differential equations (PIDE) with convolution kernel which occur naturally in various fields of science and engineering. In some cases the exact solution may be achieved. The efficiency and reliability of this method is illustrated by some examples.
On double Sumudu transform and double Laplace transform
2010
In this study, first of all, we consider wave equations with constant coefficients. By using convolution we then produce a new equation with variable coefficients. Finally, we apply two techniques: double Laplace transform and double Sumudu transform to solve the new wave equation with non-constant coefficients and establish a relationship between double Sumudu transform and double Laplace transform.
Solution of Integral Equations via Laplace ARA Transform
European Journal of Pure and Applied Mathematics
This research article demonstrates an efficient method for solving partial integro-differential equations. The intention of this research is to establish the solution of some different classes of integral equations, by utilizing the double Laplace ARA transform. We present some definitions and basic concepts related to the double Laplace ARA transform. The results of the examples support the theoretical results and show the accuracy and applicability of the presented approach.
Solving Partial Integro-Differential Equations Using Laplace Transform Method
American Journal of Computational and Applied Mathematics, 2012
Partialintegro-differential equations (PIDE) occur naturally in various fields of science, engineering and social sciences. In this article, we propose a most general form of a linear PIDE with a convolution kernel. We convert the proposed PIDE to an ordinary differential equation (ODE) using a Laplace transform (LT). Solving this ODE and applying inverse LT an exact solution of the problem is obtained. It is observed that the LT is a simple and reliable technique for solving such equations. A variety of numerical examples are presented to show the performance and accuracy of the proposed method.