Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation

Abstract

In this paper, the initial-boundary-value problems for the generalized multi-term timefractional diffusion equation over an open bounded domain G × (0, T ), G ∈ R n are considered. Based on an appropriate maximum principle that is formulated and proved in the paper, too, some a priory estimates for the solution and then its uniqueness are established. To show the existence of the solution, first a formal solution is constructed using the Fourier method of the separation of the variables. The time-dependent components of the solution are given in terms of the multinomial Mittag-Leffler function. Under certain conditions, the formal solution is shown to be a generalized solution of the initialboundary-value problem for the generalized time-fractional multi-term diffusion equation that turns out to be a classical solution under some additional conditions. Another important consequence from the maximum principle is a continuously dependence of the solution on the problem data (initial and boundary conditions and a source function) thattogether with the uniqueness and existence results -makes the problem under consideration to a well-posed problem in the Hadamard sense.

Numerical Solution of Multiterm Fractional Differential Equations Using the Matrix Mittag–Leffler Functions

“…However, even if many numerical methods for FDEs can be extended to MTFDEs, delicate issues such as numerical stability, convergence or accuracy cannot be easily predicted in this case. Many authors have worked thoroughly on their numerical solution [15][16][17][18][19][20][21][22]. We restrict our attention to the linear case that includes important models, such as the Bagley-Torvik equation [23], the fractional oscillation equation [24] and many others.…”

The Solution of Initial Boundary Value Problem with Time and Space-Fractional Diffusion Equation via a Novel Inner Product

“…There are various studies in literature supporting this conclusion [1][2][3][4][5][6][7][8]. By making use of Mittag-Leffler function, characteristic equations of fractional ODEs are solved and solutions of them are constructed efficiently.…”

Numerical Solution to the Multi-Term Time Fractional Diffusion Equation in a Finite Domain

“…(1.1). On theoretical analysis and analytical methods, we refer to Daftardar-Gejji et al [7], Luchko [19] and Jiang et al [10]. In [7], the multi-term time-fractional diffusion-wave equation with the constant coefficients was considered, and a solution of the corresponding IBV problem was represented in form of the Fourier series via the multivariate Mittag-Leffler function.…”

“…In [7], the multi-term time-fractional diffusion-wave equation with the constant coefficients was considered, and a solution of the corresponding IBV problem was represented in form of the Fourier series via the multivariate Mittag-Leffler function. In [19], a generalized multi-term time fractional diffusion equation with the variable coefficients was considered, and well-posedness of the corresponding IBV problem was proved with the help of maximum principle together with the construction of solution's representation using the Fourier method. In [10], analytical solutions of the 1D multi-term time fractional diffusion-wave/diffusion equations were obtained also using the Fourier's separating variables method.…”