Analytical solutions of some general classes of differential and integral equations by using the Laplace and Fourier transforms (original) (raw)
The Fourier–Laplace Generalized Convolutions and Applications to Integral Equations
Vietnam Journal of Mathematics, 2013
In this paper we introduce two generalized convolutions for the Fourier cosine, Fourier sine and Laplace integral transforms. Convolution properties and their applications to solving integral equations and systems of integral equations are considered. Keywords Fourier sine transform • Fourier cosine transform • Laplace transform Mathematics Subject Classification (2000) 33C10 • 44A35 • 45E10 • 45J05 • 47A30 • 47B15 1 Introduction Convolutions for integral transforms are studied in the early years of the 20th century, such as convolutions for the Fourier transform (see [2, 9, 13]), the Laplace transform (see [1, 2, 8, 13, 16-19]), the Mellin transform (see [8, 13]), the Hilbert transform (see [2, 3]), the Fourier cosine and sine transforms (see [5, 7, 13, 14]), and so on. These convolutions have many important applications in image processing, partial differential equations, integral equations, inverse heat problems (see [2-4, 8, 11-13, 15-18]).
Fourier, Laguerre, Laplace Transforms with Applications
Journal of Mathematics and Applications, 2021
In this article, the author considered certain time fractional equations using joint integral transforms. Transform method is a powerful tool for solving singular integral equations, integral equation with retarded argument, evaluation of certain integrals and solution of partial fractional differential equations. The obtained results reveal that the transform method is very convenient and effective. Illustrative examples are also provided.
New trends in Laplace type integral transforms with applications
Boletim da Sociedade Paranaense de Matemática, 2015
In this paper, the authors provided a discussion on one and two dimensional Laplace transforms and generalized Stieltjes transform and their applications in evaluating special series and integrals. Finally, we implemented the joint Laplace – Fourier transforms to construct exact solution for a variant of the Kd.V equation. Illustrative examples are also provided.
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.
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.
Application of the Fourier Transform to the Solution of Singular Integral Equations
Mathematical Notes, 2003
A method for solving a certain system of singular integral equations with constant coefficients is proposed. It is based on a procedure for reducing singular equations to equations with continuous difference kernel; the solution of the latter is constructed by using the classical Fourier transform in the class of absolutely integrable functions. Explicit expressions for the solution of the singular integral equations under consideration are obtained.
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.