Multi-term Time Fractional Differential Equations Research Articles

Lie symmetry group, exact solutions and conservation laws for multi-term time fractional differential equations

<abstract><p>In this paper, the time fractional Benjamin-Bona-Mahony-Peregrine (BBMP) equation and time-fractional Novikov equation with the Riemann-Liouville derivative are investigated through the use of Lie symmetry analysis and the new Noether's theorem. Then, we construct their group-invariant solutions by means of Lie symmetry reduction. In addition, the power-series solutions are also obtained with the help of the Erdélyi-Kober (E-K) fractional differential operator. Furthermore, the conservation laws for the time-fractional BBMP equation are established by utilizing the new Noether's theorem.</p></abstract>

Inverse Problem for a Multi-Term Fractional Differential Equation

Abstract Inverse problem for a family of multi-term time fractional differential equation with non-local boundary conditions is studied. The spectral operator of the considered problem is non-self-adjoint and a bi-orthogonal set of functions is used to construct the solution. The operational calculus approach has been used to obtain the solution of the multi-term time fractional differential equations. Integral type over-determination condition is considered for unique solvability of the inverse problem. Different estimates of multinomial Mittag-Leffler functions alongside Banach fixed point theorem are used to prove the unique local existence of the solution of the inverse problem. Stability of the solution of the inverse problem on the given datum is established.

Convergence of Finite Difference Method in Positive Time for Multi-Term Time Fractional Differential Equations

Convergence of Finite Difference Method in Positive Time for Multi-Term Time Fractional Differential Equations

Operational shifted hybrid Gegenbauer functions method d for solving multi-term time fractional differential equations

In this paper, we propose an efficient operational formulation of spectral tau method d for solving multi-term time fractional differential equations with initial-b boundary conditions. The shifted hybrid Gegenbauer functions (SHGFs) operational matrices of Riemann-Liouville fractional integral and Caputo fractional derivatives are presented. By using these operational matrices, the shifted hybrid Gegenbauer functions tau method for both temp oral and spatial discretization are presented, which allow us to introduce an efficient spectral method for solving such problems. Finally, numerical results show good deal with the theoretical analysis.

Analytical Solutions for Multi-Term Time-Space Fractional Partial Differential Equations with Nonlocal Damping Terms

AbstractIn this paper, we consider the analytical solutions of multi-term time-space fractional partial differential equations with nonlocal damping terms for general mixed Robin boundary conditions on a finite domain. Firstly, method of reduction to integral equations is used to obtain the analytical solutions of multi-term time fractional differential equations with integral terms. Then, the technique of spectral representation of the fractional Laplacian operator is used to convert the multi-term time-space fractional partial differential equations with nonlocal damping terms to the multi-term time fractional differential equations with integral terms. By applying the obtained analytical solutions to the resulting multi-term time fractional differential equations with integral terms, the desired analytical solutions of the multi-term time-space fractional partial differential equations with nonlocal damping terms are given. Our results are applied to derive the analytical solutions of some special cases to demonstrate their applicability.

Analytical solutions of multi-term time fractional differential equations and application to unsteady flows of generalized viscoelastic fluid

This paper derives analytical solutions for a class of new multi-term fractional-order partial differential equations, which include the terms for spatial diffusion, time-fractional diffusion (multi-term) and reaction. These models can be used to describe the nonlinear relationship between the shear stress and shear rate of generalized viscoelastic Oldroyd-B fluid and Burgers fluid. By using a modified separation of variables method, the governing fractional-order partial differential equations are transformed into a set of fractional-order ordinary differential equations. Mikusiński-type operational calculus is then employed to obtain the exact solutions of the linear fractional ordinary differential equations with constant coefficients. The solutions are expressed in terms of multivariate Mittag-Leffler functions. Different situations for the unsteady flows of generalized Oldroyd-B fluid and Burgers fluid due to a moving plate are considered via examples. Integral representations of the solutions are presented. It is shown that the presented results reduce to the corresponding results for classical Navier–Stokes, Oldroyd-B, Maxwell and second-grade fluids as special cases.

Analytical solutions for the multi-term time-fractional diffusion-wave/diffusion equations in a finite domain

Multi-term time-fractional differential equations have been used for describing important physical phenomena. However, studies of the multi-term time-fractional partial differential equations with three kinds of nonhomogeneous boundary conditions are still limited.In this paper, a method of separating variables is used to solve the multi-term time-fractional diffusion-wave equation and the multi-term time-fractional diffusion equation in a finite domain. In the two equations, the time-fractional derivative is defined in the Caputo sense. We discuss and derive the analytical solutions of the two equations with three kinds of nonhomogeneous boundary conditions, namely, Dirichlet, Neumann and Robin conditions, respectively.