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.
Highlights
In this paper, we will consider the following high (d-) dimensional parabolic Volterra integro-differential equation with memory kernel K(t) in the infinite domain [1]: ∂u(x, t) + ∂t Z tK(t − t0 )u(x, t0 )dt0 = ∆u(x, t) + f (x, t), (1)which satisfies the initial condition and boundary conditions below, u(x, 0) = g(x), lim u(x, t) = 0, |x|→∞t > 0, x = ( x1, x2, · · ·, x d ) ∈ R d
To the authors’ knowledge, there are no studies on analytical solutions of parabolic partial Volterra integro-differential equation in the infinite domain
We present some fundamental definitions and lemmas that are used throughout this paper
Summary
We will consider the following high (d-) dimensional parabolic Volterra integro-differential equation with memory kernel K(t) in the infinite domain [1]:. Proposed the artificial boundary method to solve parabolic Volterra integro differential equations (one/two-dimensional) in infinite spatial domains. Volterra integro-differential equation in one-dimensional finite and infinite spatial domains using spectral collocation methods. To the authors’ knowledge, there are no studies on analytical solutions of parabolic partial Volterra integro-differential equation in the infinite domain. Our goal is mainly to discuss analytical solutions of Equation (1) with three different kinds of memory kernel functions in the infinite domain. In. Section 3, the analytical solutions of parabolic Volterra integro-differential equation with three different kinds of memory kernel are demonstrated in the infinite domain.
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