Finite Difference and Spline Approximation for Solving Fractional Stochastic Advection-Diffusion Equation
- PDF / 759,771 Bytes
- 11 Pages / 595.276 x 790.866 pts Page_size
- 56 Downloads / 188 Views
(0123456789().,-volV)(0123456789(). ,- volV)
RESEARCH PAPER
Finite Difference and Spline Approximation for Solving Fractional Stochastic Advection-Diffusion Equation Farshid Mirzaee1
•
Khosro Sayevand1 • Shadi Rezaei1 • Nasrin Samadyar1
Received: 2 April 2020 / Accepted: 11 November 2020 Ó Shiraz University 2020
Abstract This paper is concerned with numerical solution of time fractional stochastic advection-diffusion type equation where the first order derivative is substituted by a Caputo fractional derivative of order a (0\a 1). This type of equations due to randomness can rarely be solved, exactly. In this paper, a new approach based on finite difference method and spline approximation is employed to solve time fractional stochastic advection-diffusion type equation, numerically. After implementation of proposed method, the under consideration equation is transformed to a system of second order differential equations with appropriate boundary conditions. Then, using a suitable numerical method such as the backward differentiation formula, the resulting system can be solved. In addition, the error analysis is shown in some mild conditions by ignoring the error terms OðDt2 Þ in the system. In order to show the pertinent features of the suggested algorithm such as accuracy, efficiency and reliability, some test problems are included. Comparison achieved results via proposed scheme in the case of classical stochastic advection-diffusion equation (a ¼ 1) with obtained results via wavelets Galerkin method and obtained results for other values of a with the values of exact solution confirm the validity, efficiency and applicability of the proposed method. Keywords Fractional stochastic advection-diffusion equation Stochastic partial differential equations Caputo fractional derivative Finite difference method Spline approximation Brownian motion process Mathematics Subject Classification 60H15 35R11 26A33 65M20 41A15
1 Introduction Since, there exist connection between phenomena in the real world and partial differential equations (PDEs), so they have been extensively used to formulate a wide range of applied problems. In many practical situations such as advection-diffusion equation which is one of the most important PDEs and arising in ground water flows, such & Farshid Mirzaee [email protected]; [email protected] Khosro Sayevand [email protected] Shadi Rezaei [email protected] Nasrin Samadyar [email protected] 1
Faculty of Mathematical Sciences and Statistics, Malayer University, P. O. Box 65719-95863, Malayer, Iran
ideal information is rarely encountered. For example, our information about the permeability of the soil, magnitude of source term, inflow or outflow conditions, etc, are not accurate. Uncertainties in this problem can be modeled by stochastic advection-diffusion equation. Over a long period of time, these random factors were ignored due to lack of powerful computational tools and problems were modeled by deterministic advection-
Data Loading...
