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A numerical scheme for time-fractional fourth-order reaction-diffusion model

Koç Dilara Altan

Journal of Applied Mathematics and Computational Mechanics

2023

Vol. 22, nr 2

15--25

In fractional calculus, the fractional differential equation is physically and theoretically important. In this article an efficient numerical process has been developed. Numerical solutions of the time fractional fourth order reaction diffusion equation in the sense of Caputo derivative is obtained by using the implicit method, which is a finite difference method and is developed by increasing the number of iterations. The advantage of the implicit difference scheme is unconditionally stable. The stability analysis and convergency have been proven. A numerical example has been presented, and the validity of the method is supported by tables and graphics.

Implicit finite difference method for the space fractional heat conduction equation with the mixed boundary condition

Brociek R., Słota D.

Silesian Journal of Pure and Applied Mathematics

2016

Vol. 6, iss. 1

125--136

This paper presents the numerical solution of the space fractional heat conduction equation with Neumann and Robin boundary conditions. In described equation the Riemann-Liouville fractional derivative is used. Considered model is solved by using the implicit ﬁnite diﬀerence method. The paper also presents the numerical examples to illustrate the accuracy and stability of described method.

Parallel computations of the step response of a floor heater with the use of a graphics processing unit. Part 1: models and algorithms

Gołębiowski J., Forenc J.

Bulletin of the Polish Academy of Sciences. Technical Sciences

2013

Vol. 61, nr 4

943--948

The article presents a method of computing the step response of an air floor heater. The method implements parallel algorithms on a graphics processing unit. In the analyzed concrete slab heating ducts are placed. Hot air is transferred through them, thanks to which the heat penetrates into the slab. Heat transfer into the environment takes place on the top surface of the floor by natural convection and radiation. The bottom surface of the slab is thermally insulated. A two-dimensional heat equation was discretized with the use of the implicit finite difference method. In order to solve the obtained system of equations, the conjugate gradient method was used. Moreover, in order to examine the possibility of shortening the computations time, the algorithm of this method was implemented on a graphics processing unit. A computer program, using the CUDA parallel computing platform and linear algebra libraries CUBLAS and CUSPARSE, was developed.