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Integral and Integro-Differential Equations
Published in K.T. Chau, Theory of Differential Equations in Engineering and Mechanics, 2017
In the first part of this chapter, we will summarize the idea of integral transforms versus eigenfunction expansion that we discussed in the last chapter. Various types of integral transforms will be introduced, including the Fourier transform, Hankel transform, Mellin transform, Hilbert transform, and Laplace transform. When integral transforms are applied to ordinary differential equations, ODEs will become algebraic equations that can be solved readily most of the time. If integral transforms are applied to partial differential equations of two variables, the PDE will become an ODE. The resulting ODE is, of course, much easier to solve than the original PDE. For PDEs with variables, each time an integral transform is applied, the resulting PDE will only involve n - 1 variables. Therefore, when an integral transform is applied repeatedly, the PDE will eventually become an algebraic equation. However, due to space limitations we only cover the basics of integral transform and its introduction is primarily for setting the scene for the more difficult problems of integral equations and of integro-differential equations.
Transform Techniques in Physics
Published in Russell L. Herman, A Course in Mathematical Methods for Physicists, 2013
The Fourier and Laplace transforms are examples of a broader class of transforms known as integral transforms. For a function f (x) defined on an interval (a, b), we define the integral transform F(k)=∫abK(x,k)f(x)dx,Integral transform on [a, b] with respect to the integral kernel, K(x, k).
Properties of Laplace transforms
Published in John Bird, Bird's Higher Engineering Mathematics, 2021
As stated in the preceding chapter, the Laplace transform is a widely used integral transform with many applications in engineering, where it is used for analysis of linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices and mechanical systems. The Laplace transform is also a valuable tool in solving differential equations, such as in electronic circuits, and in feedback control systems, such as in stability and control of aircraft systems. This chapter considers further transforms together with the Laplace transform of derivatives that are needed when solving differential equations.
Fractional effects on solitons in a 1D array of rectangular ferroelectric nanoparticles
Published in Waves in Random and Complex Media, 2020
Yaouba Amadou, Mibaile Justin, Malwe B. Hubert, Gambo Betchewe, Serge Y. Doka, Kofane T. Crepin
Hence, Jumarie [6,7] suggested a modification of the Riemann–Liouville fractional derivative, where f is a continuous function which reads The Jumarie's fractional derivative has the following properties [13]: The local fractional derivatives introduced in physical models can describe sound attenuation in complex media. From these considerations, one can convert fractional derivative in the sense of the Jumarie's modified Riemann–Liouville derivative to integer order partners by means of the fractional complex transform. Therefore, transforms are useful tools in solving problems of applied sciences. To mention a few, we have the Laplace transform, the Fourier transform, the wave transformation, the Backlung transformation, the integral transform, etc. Though, many applications of the fractional complex transform appeared in literatures, it is observed that the previous merits on transforms without fractional derivative can be completely eliminated when the local fractional derivative is used. For example, consider a local fractional partial differential equation in the form [30] By means of the fractional complex transform we obtain
The negative exponential transformation: a linear algebraic approach to the Laplace transform
Published in International Journal of Mathematical Education in Science and Technology, 2023
The real Laplace transform is first encountered in the second year of the undergraduate curriculum of South African universities primarily as a tool to solve ordinary differential equations (see Section 2). The Laplace transform is a versatile integral transform used widely in physics, electrical engineering, control engineering and signal processing. In signal processing, the Laplace transform is a transformation from the time domain where the input signal are functions of time to the frequency domain where the inputs are now functions of complex angular frequency. We will address the signal processing approach of the Laplace transform as functions of angular frequency as poles (see Example 4.1).
A review of hybrid integral transform solutions in fluid flow problems with heat or mass transfer and under Navier–Stokes equations formulation
Published in Numerical Heat Transfer, Part B: Fundamentals, 2019
Renato M. Cotta, Kleber M. Lisboa, Marcos F. Curi, Stavroula Balabani, João N. N. Quaresma, Jesus S. Perez-Guerrero, Emanuel N. Macêdo, Nelson S. Amorim
Integral transforms have been widely employed in the solution of differential equations, though their usefulness is not limited to this purpose. It is recognized that the Leonhard Euler has in fact introduced the concept of integral transforms in handling second-order differential equations [1], first in 1763 for a specific differential equation, and later on in 1769 when the treatment was more systematic and complete [1]. On the other hand, Fourier in his 1822 treatise [2] advanced the idea of Separation of Variables, so as to handle and interpret the solutions of the newly derived heat conduction equation, after proposing the constitutive equation known as Fourier’s law. His work provided not only the modern mathematical theory of heat conduction but also introduced the well-known Fourier series and Fourier transforms. However, it appears that it was in the work of Acad. N.S. Koshlyakov [3], which the integral transform method gained a more general formalism based on eigenfunction expansions and was first extended to handle linear partial differential equations in finite media with nonhomogeneous terms, either on the main equation or in the boundary conditions, as described in his textbook [4]. This concept of a more general integral transform approach based on eigenfunctions from Sturm–Liouville eigenvalue problems was further explored by Koshlyakov and coworkers [5] and Eringen [6], among others. In the 60s, Luikov [7], Mikhailov [8], and Ozisik [9] made some of the most fundamental contributions for the full establishment of this analytical approach in the heat and mass transfer field. The consolidation of these ideas was systematically presented in the compendium of Mikhailov and Ozisik [10] of 1984, which organized the integral transform analysis of heat and mass diffusion into seven fairly wide classes of linear problems.