An Inverse Source Problem in Time-Space Fractional Differential Equations

In this paper we study a one dimensional inverse source problem. Distinguishability according to the source function in time-space fractional equation Db t u(x, t) = D2a x u(x, t)+ f (x), 1

Inverse Problem for a Time Fractional Parabolic Equation with Nonlocal Boundary Conditions

This article considers an inverse problem of time fractional parabolic partial differential equations with the nonlocal boundary condition. Dirichlet-measured output data are used to distinguish the unknown coefficient. A finite difference scheme is constructed and a numerical approximation is made. Examples and numerical experiments, such as man-made noise, are provided to show the stability and efficiency of this numerical method

A mathematical model and numerical solution of interface problems for steady state heat conduction

We study interface (or transmission) problems arising in the steady state heat conduction for layered medium. These problems are related to the elliptic equation of the form Au:=−∇(k(x)∇u(x))=F(x), x∈Ω⊂ℝ2, with discontinuous coefficient k=k(x). We analyse two types of jump (or contact) conditions across the interfaces Γδ−=Ω1∩Ωδ and Γδ+=Ωδ∩Ω2 of the layered medium Ω:=Ω1∪Ωδ∪Ω2. An asymptotic analysis of the interface problem is derived for the case when the thickness (2δ>0) of the layer (isolation) Ωδ tends to zero. For each case, the local truncation errors of the used conservative finite difference scheme are estimated on the nonuniform grid. A fast direct solver has been applied for the interface problems with piecewise constant but discontinuous coefficient k=k(x). The presented numerical results illustrate high accuracy and show applicability of the given approach

On Fractional Newton-Type Method for Nonlinear Problems

The current manuscript is concerned with the development of the Newton–Raphson method, playing a significant role in mathematics and various other disciplines such as optimization, by using fractional derivatives and fractional Taylor series expansion. The development and modification of the Newton–Raphson method allow us to establish two new methods, which are called first- and second-order fractional Newton–Raphson (FNR) methods. We provide convergence analysis of first- and second-order fractional methods and give a general condition for the convergence of higher-order FNR. Finally, some illustrative examples are considered to confirm the accuracy and effectiveness of both methods

Inverse Problem for a Time Fractional Parabolic Equation with Nonlocal Boundary Conditions

This article considers an inverse problem of time fractional parabolic partial differential equations with the nonlocal boundary condition. Dirichlet-measured output data are used to distinguish the unknown coefficient. A finite difference scheme is constructed and a numerical approximation is made. Examples and numerical experiments, such as man-made noise, are provided to show the stability and efficiency of this numerical method