Finite Fourier transform for solving potential and steady-state temperature problems
- Kamel Al-Khaled^{1}Email author
https://doi.org/10.1186/s13662-018-1552-8
© The Author(s) 2018
Received: 10 January 2018
Accepted: 9 March 2018
Published: 20 March 2018
Abstract
The derivation of this paper is devoted to describing the operational properties of the finite Fourier transform method, with the purpose of acquiring a sufficient theory to enable us to follow the solutions of boundary value problems of partial differential equations, which has some applications on potential and steady-state temperature. Numerical calculations show that the present method gives higher accuracy with less computation time than other, traditional methods, like the finite difference method.
Keywords
1 Introduction
Many boundary value problems can be solved by means of integral transformations, such as the Laplace transform function, which transform a differential equation into an algebraic equation in which the boundary conditions are automatically considered. After solving the algebraic equation, one finds the solution of the original equation by means of the inverse transformation. Similarly, partial differential equations are changed into ordinary differential equations by applying these transformations. Two transformations which are particularly useful in solving boundary value problems are the finite Fourier sine and cosine transformations.
The particular transformation discussed in this paper is the finite Fourier transform, which is applicable to equations in which only the even order derivatives (of the function) with respect to transformed variable will be treated. The finite Fourier transform method is one of various analytical techniques in which exact solutions of boundary value problems can be constructed. The transform exists for all bounded, piecewise continuous functions over a finite interval. In recent years, the finite Fourier transform method has been applied to a wide class of boundary value problems in many interesting mathematics, physics, chemistry and engineering areas [1–3]. Many other transforms exist which may be used to solve PDEs [4, 5].
A feature which makes the finite transform a very economical method, is that the inverse transform may be solved only for regions of interest [6–8]. The most widely used methods for the solution of boundary value problems are based on finite differences. These methods require certain assumptions about where the finite difference equals the derivative which by necessity have to be most loosely made on the boundaries. One must form many grid points near the boundary in order for the numerical solution to be accurate. Another feature of the finite Fourier transform method is that it gives the exact solution at the boundary [9].
This paper is written in two parts. Sect. 2 will be devoted to a description of the operational properties of these transformations with the purpose of acquainting the reader with sufficient theory to enable us to follow the solutions of the problems given in Sect. 3, which consists of a group of simple boundary value problems and their solutions.
2 Basic definitions
Definition 1
Let \(F(x)\) be a function that is sectionally continuous in the interval \(0< x<\pi\), the finite Fourier sine and cosine transformations are defined as follows:
2.1 Some operational properties
2.2 Convolution
Theorem 2.1
Proof
3 Solutions of partial differential equations
3.1 Vibrations of a horizontal string with fixed ends
3.2 Transverse vibrations of a beam
3.3 Applications to problems in heat conduction
Application 1
Application 2
Application 3
Application 4
4 Concluding remarks
Boundary value problems of partial differential equations concerned with temperature as the unknown may be solved by a finite Fourier transform method. The temperature at points other than the boundary, if they should be needed, can be obtained by summing the Fourier coefficients. For potential problems, the temperature at the boundary should be as accurate as possible. The finite Fourier transform method which gives the exact boundary temperature within the computer accuracy is shown to be an extremely powerful mathematical tool for the analysis of boundary value problems of partial differential equations with applications in physics. Also the finite Fourier transform method differs from the usual Fourier transformation method in that the solutions are obtained without performing the inverse Fourier transforms. In principle, the finite Fourier transform method may be extended to analog simulations of heat equations in three space variables, and it may also be a very efficient technique for the solution of multidimensional heat equations.
Declarations
Acknowledgements
The author is grateful to the editor and the reviewers for their careful reading and useful comments. This research was partially supported by Jordan University of Science and Technology.
Authors’ contributions
The author wrote this paper, read and approved the final manuscript.
Competing interests
The author declares that he has no competing interest.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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