Abstract
In this paper, an initial-boundary value problem for a one-dimensional linear time-dependent fractional Schrödinger pseudoparabolic partial differential equation with Caputo derivative of order α ∈ 0,1 is being considered. Two strong numerical methods are employed to acquire the solutions to the problem. The first method used is the double Laplace decomposition method where closed-form solutions are obtained for any α ∈ 0,1 . As the second method, the implicit finite difference scheme is applied to obtain the approximate solutions. To clarify the performance of these two methods, numerical results are presented. The stability of the problem is also investigated.
Highlights
Double Laplace Decomposition MethodWe give some basic definitions and theorems of the double Laplace transform
With the same initial-boundary conditions mentioned for one-dimensional parabolic partial differential equation [1]: ut(x, t) kuxx(x, t)
pseudoparabolic partial differential equation (PPPDE) appear in several fields of mechanics and physics. ey are used to investigate homogeneous fluid flow in fractured rocks, quasi-stationary paths in semiconductors, thermodynamic and transportation phenomena, and different physical systems
Summary
We give some basic definitions and theorems of the double Laplace transform. E double Laplace transform of the function u(x, t) is defined by the following double improper integral:. For a positive constant M, the double Laplace transform of u(x, t) exists for all p and s provided Re(p) > e1 and Re(s) > e2. K, exist, the double Laplace respect to the time variable is defined as. Applying the double Laplace transform with respect to x and t to equation (11) yields (13). Taking the inverse double Laplace transform to both sides of equation (15), it results in u(x, t). Stability can be acquired by applying the method of analyzing the eigenvalues of the iteration matrices of the schemes. FDS (26) is conditionally stable subject to the following space step limitation:. As we know that θni,i ρ(θn) for 2 ≤ i ≤ N + 1 and from the induction hypothesis, we conclude that ρ(θn+1) < 1 subject to condition (35). us, we acquire the required result by induction
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