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

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.


Introduction
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]: which satisfies the initial condition and boundary conditions below, where function f (x, t) and memory kernel function K(t) are assumed to be sufficiently smooth functions, and ∆ is the d-dimensional Laplacian operator. Parabolic Volterra integro-differential equations present an important model to simulate the effects of the "memory" of the system. The model is based on a partial differential equation, combining the partial differentiation and integral term containing the unknown kernel function that leads to parabolic Volterra integro-differential equations. In general, the model nonlocal effects of the memory of past history cannot be described by partial differential equations. Therefore, more and more researchers have studied the solution of integro-differential equations.
Diffusion is one of the most important and ubiquitous phenomena in nature. The parabolic Volterra integro-differential equation model Equation (1) can be also describe anomalous diffusion; see references [2,3]. The reaction diffusion level most depends on the choice of the kernel K(t) present in the nonlocal reaction diffusion (integral) term.
In this section, we will introduce some fundamental lemmas for the Laplace transform (L), which will be help us handle some problems in the next section. Lemma 1. Let s, b, α, λ n ∈ R + , and then the following inverse Laplace transform (L −1 ) is true [18].
It is worth noting that in case λ = 0, the structure of Lemma 3 is equivalent to Lemma 2. Lemma 4 gives one important d-dimensional integral formula for the Mittag-Leffler function.

Lemma 4.
For arbitrary α > 0, β is an arbitrary complex number, µ > 0, and a ∈ R, establishing the following formula: [20] R d In fact, here J d/2−1 (·) is a Bessel function, and E (n) α,β (z) denotes the n-th derivatives of the two-parameter Mittag-Leffler function. n-th derivatives of the two-parameter Mittag-Leffler function can be expressed in terms of the Fox-H function as Furthermore, on the right of Equation (10) is the following: where H 1,0 0,2 (z) [15] is the Fox-H function. (1) can be written as follows:
Theorem 1. The analytical solution of parabolic Volterra integro-differential Equation (11) with boundary conditions and initial condition (2) can be expressed as the following analytical form In which G(x, t) is the Green function and reads as In general, this denotes Proof. This denotes F {u(x, t)} := u(k, t) as the Fourier transform of u(x, t) with respect to variable x, and L {u(x, t)} :=û(x, s) as the Laplace transform of u(x, t) with respect to variable t. Taking the Laplace transform with respect to the time variable t and the Fourier transform with respect to the spatial variable x to Equation (11), we obtain a nonhomogeneous equation, which can be written as the following: Using the initial condition, it yields: Employing Lemma 2, we can get Substituting Equations (15) and (16) into the first term of Equation (14), we have By applying inverse Laplace transform (L −1 ), the convolution definition of a Laplace transform and Definition 2, we can obtain the inverse Laplace transform of the first term in Equation (14) as follows: It follows that the inverse Laplace transform of the second term in Equation (14) is According to Equations (18) and (19), we can obtain u(k, t) from Equation (14) u(k, t) = Equation (20) can be further manipulated by employing an inverse Fourier transform: We need 1 τ αδ and we presented that the series and integrals (21) are convergent. And E The second term in Equation (21) can be further manipulated as follows Denote Green function G(x, t) is Therefore, we complete the proof of Theorem 1. (1) can be written as the following:

Analytical Solution with a Frictional Memory Kernel of Power
Theorem 2. The analytical solution of parabolic Volterra integro-differential Equation (23) with boundary conditions and initial condition (2) can be expressed as The Green function G(x, t) is given as Remark. Employing the properties of the Fox-H functions, Green function G(x, t) can be expressed as a power series expansion [15] G( . For |x| 2t 1 2 1, therefore, the power-law asymptotics behavior is given by .
Proof. Employing the Laplace transform with respect to variable t and Fourier transform with respect to variable x, respectively. We obtain the following nonhomogeneous equation Taking into account the initial condition, we obtain the following nonhomogeneous differential equation, and Equation (25) can be rewritten aŝ Using the technique introduced by [18], we have Expanding the third section on the right of Equation (27) and simplifying, one can easily get Employing Lemma 2 in Equation (28), the first term of Equation (26) can be expressed as We need < 1, and showed that series (29) is convergent. Also, E n+1 2−α,n+1 (·) is a generalization of the three-parameter Mittag-Leffler function. According to the three-parameter Mittag-Leffler function definition, from Equation (29), we get Applying the convolution property of the Laplace transform and integral operator E w;γ,κ a+;α,β ϕ Definition 2, the inverse Laplace transform of the first term in Equation (26) can be obtained as follows: Considering the relationship between the generalized Mittag-Leffler function and the Fox-H function, the inverse Laplace transform of the second term in Equation (26) can be expressed as [15] L −1 1 and Using the inverse Fourier transform (F −1 ) in Equation (33), we can obtain Using Lemma 4, we get the following result from Equation (35) h(x, t) = 1 Employing a Hankel transform and the properties of the Fox-H functions [15,21,22], Equation (36) can be written as Substituting Equation (37) into the inverse Fourier transform of Equation (34), we can obtain Then Equation (32) can also be written formally as Applying an inverse Laplace transform to Equation (26), we can finally find Equation (39) can be further manipulated by employing inverse Fourier transform and Fourier convolution theorem, respectively. Accordingly, the Theorem 2 is clearly demonstrated.
3.3. Analytical Solution with Frictional Memory Kernel of Exponential Factor Type K(t) = t β e −λt , β > −1 In this case, Equation (1) can be written in the following form Theorem 3. The analytical solution of parabolic Volterra integro-differential Equation (40) with boundary conditions and initial condition (2) can be expressed as the following analysis formula: Letting k β = Γ(β + 1), using the asterisk ( * ) to denote a Laplace convolution, the Green function G(x, t) is given by Proof. Applying the Laplace and Fourier transform with respect to the time variable t and spatial variable x to Equation (40), and using the initial conditions, Equation (40) can be written as From Equation (42), we havê u(k, s) = 1 Applying power series expansion, one obtains Combining with Lemma 3, Equation (44) can be expressed as We need (|k| 2 −λ)(s+λ) β+1 (s+λ) β+2 +k β < 1, and showed that series (45) is convergent. In addition, E n+1 β+2,n+1 (z) is a generalization of the Mittag-Leffler function.
Through Equation (45), the first term in Equation (43) can be formally written as Finally, the inverse Laplace transform of the first term in Equation (43) can be rewritten as After analogous manipulation, the inverse Laplace transform of the second term of Equation (43) can be expressed as Performing an inverse Laplace transform in Equation (43), we finally get The solution is now obtained by performing an inverse Fourier transform in Equation (49), which produces where the Green function is denoted as Therefore, we complete the proof of Theorem 3.
(1 / 2,1 / 2), The graphical representation of solution (52) for the different parameters of α and t are plotted in Figures 1-4, respectively. (a) Times is fixed, the peak of curves will decrease gradually with the increasing of parameter α . (b) The peak of curves distinct decrease and get fit with the increasing of parameter α under given time.

Conclusions
In practical applications, different types of the frictional memory kernel ( ) t  have been used to describe a wide variety of complex dynamics and physical phenomena with memory effects. In this paper, by applying the Laplace transform method to the time variable and the Fourier transform to the spatial variable, we obtained analytical solutions to parabolic Volterra integro-differential equations with three different kinds of memory kernel in the infinite domain.
The analytical solutions to the parabolic Volterra integro-differential equation consist of some special functions, such as the multi-parameter Mittag-Leffler function and Fox-H function. It is worth mentioning that the analytical solution provided in Equation (24) can also be obtained by applying the references method [23][24][25] to Equation (23). In the end, some curves of the analytical solution are given, and the curves exhibit a long tail behavior for a large time. We found that a parabolic Volterra integro-differential equation with a power-law memory kernel is characterized by anomalous diffusion. The analytic solution of (1) can be found in some special cases, but in general it is difficult to obtain because of the non-local property of the integral term. Thus, in many cases, the more reasonable option is to find its numerical solution. Meanwhile, the analytical solutions we obtained from parabolic Volterra integro-differential equations with different types of frictional memory kernel provide great convenience for practical applications. (a) Times is fixed, the peak of curves will decrease gradually with the increasing of parameter α . (b) The peak of curves distinct decrease and get fit with the increasing of parameter α under given time.

Conclusions
In practical applications, different types of the frictional memory kernel ( ) t  have been used to describe a wide variety of complex dynamics and physical phenomena with memory effects. In this paper, by applying the Laplace transform method to the time variable and the Fourier transform to the spatial variable, we obtained analytical solutions to parabolic Volterra integro-differential equations with three different kinds of memory kernel in the infinite domain. The analytical solutions to the parabolic Volterra integro-differential equation consist of some special functions, such as the multi-parameter Mittag-Leffler function and Fox-H function. It is worth mentioning that the analytical solution provided in Equation (24) can also be obtained by applying the references method [23][24][25] to Equation (23). In the end, some curves of the analytical solution are given, and the curves exhibit a long tail behavior for a large time. We found that a parabolic Volterra integro-differential equation with a power-law memory kernel is characterized by anomalous diffusion. The analytic solution of (1) can be found in some special cases, but in general it is difficult to obtain because of the non-local property of the integral term. Thus, in many cases, the more reasonable option is to find its numerical solution. Meanwhile, the analytical solutions we obtained from parabolic Volterra integro-differential equations with different types of frictional memory kernel provide great convenience for practical applications.

Conclusions
In practical applications, different types of the frictional memory kernel K(t) have been used to describe a wide variety of complex dynamics and physical phenomena with memory effects. In this paper, by applying the Laplace transform method to the time variable and the Fourier transform to the spatial variable, we obtained analytical solutions to parabolic Volterra integro-differential equations with three different kinds of memory kernel in the infinite domain. The analytical solutions to the parabolic Volterra integro-differential equation consist of some special functions, such as the multi-parameter Mittag-Leffler function and Fox-H function. It is worth mentioning that the analytical solution provided in Equation (24) can also be obtained by applying the references method [23][24][25] to Equation (23). In the end, some curves of the analytical solution are given, and the curves exhibit a long tail behavior for a large time. We found that a parabolic Volterra integro-differential equation with a power-law memory kernel is characterized by anomalous diffusion. The analytic solution of (1) can be found in some special cases, but in general it is difficult to obtain because of the non-local property of the integral term. Thus, in many cases, the more reasonable option is to find its numerical solution. Meanwhile, the analytical solutions we obtained from parabolic Volterra integro-differential equations with different types of frictional memory kernel provide great convenience for practical applications.