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Entropy 2016, 18(10), 344; doi:10.3390/e18100344

The Analytical Solution of Parabolic Volterra Integro-Differential Equations in the Infinite Domain

Department of Applied Mathematics, School of sciences, Xi’an University of Technology, Xi’an 710054, China
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Academic Editor: Angelo Plastino
Received: 11 August 2016 / Revised: 14 September 2016 / Accepted: 19 September 2016 / Published: 23 September 2016
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Abstract

This article focuses on obtaining analytical solutions for d-dimensional, parabolic Volterra integro-differential equations with different types of frictional memory kernel. Based on Laplace transform and Fourier transform theories, the properties of the Fox-H function and convolution theorem, analytical solutions for the equations in the infinite domain are derived under three frictional memory kernel functions. The analytical solutions are expressed by infinite series, the generalized multi-parameter Mittag-Leffler function, the Fox-H function and the convolution form of the Fourier transform. In addition, graphical representations of the analytical solution under different parameters are given for one-dimensional parabolic Volterra integro-differential equations with a power-law memory kernel. It can be seen that the solution curves are subject to Gaussian decay at any given moment. View Full-Text
Keywords: parabolic Volterra integro-differential equations; memory kernel; Laplace transform; Fourier transform; convolution theorem; analytical solution parabolic Volterra integro-differential equations; memory kernel; Laplace transform; Fourier transform; convolution theorem; analytical solution
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Zhao, Y.; Zhao, F. The Analytical Solution of Parabolic Volterra Integro-Differential Equations in the Infinite Domain. Entropy 2016, 18, 344.

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