Shahid Chamran University of AhvazJournal of Advanced Mathematical Modeling2251-808812220220622Numerical investigation of a new difference scheme on a graded mesh for solving the time-space fractional sub-diffusion equations with nonsmooth solutionsNumerical investigation of a new difference scheme on a graded mesh for solving the time-space fractional sub-diffusion equations with nonsmooth solutions2122311765110.22055/jamm.2022.39099.1977FAMojtabaFardiDepartment of Applied Mathematics, Faculty of Mathematical Science, Shahrekord University,
Shahrekord, P. O. Box 115, Iran.ٍٍEbrahimAminiDepartment of Mathematics, Payme Noor University, P. O. Box 19395-4697 Tehran, IRAN.Journal Article20211105In this paper, we provide a new difference scheme on a graded mesh for solving the time-space fractional diffusion problem. In this equation the time derivative is the Caputo of order $gammain(0,1)$ and the space derivative is the Riesz of order $alphain(1,2]$. The stability and convergence of the difference scheme are discussed which provides the theoretical basis of the proposed schemes. We prove that the new difference scheme is unconditionally stable. Also, we find that the difference scheme is convergent with order $min{2-gamma,rgamma}$ in time for all $gammain (0,1)$ and $alpha in (1,2]$. A test example is given to verify the efficiency and accuracy of the difference scheme.In this paper, we provide a new difference scheme on a graded mesh for solving the time-space fractional diffusion problem. In this equation the time derivative is the Caputo of order $gammain(0,1)$ and the space derivative is the Riesz of order $alphain(1,2]$. The stability and convergence of the difference scheme are discussed which provides the theoretical basis of the proposed schemes. We prove that the new difference scheme is unconditionally stable. Also, we find that the difference scheme is convergent with order $min{2-gamma,rgamma}$ in time for all $gammain (0,1)$ and $alpha in (1,2]$. A test example is given to verify the efficiency and accuracy of the difference scheme.https://jamm.scu.ac.ir/article_17651_ddb3bf3274be1ad625f7a3480b527ac9.pdf