In this paper, we consider a type of fractional diffusion equation (FDE) with variable coefficient on a finite domain. Firstly, we utilize a second-order scheme to approximate the Riemann-Liouville fractional derivative and present the finite difference scheme. Specifically, we discuss the Crank-Nicolson scheme and solve it in matrix form. Secondly, we prove the stability and convergence of the...

In this article, we verify the existence and uniqueness of a positive and nondecreasing solution for nonlinear boundary value problem of fractional differential equation in the form Dα 0+ x(t) + f(t, x(t)) = 0, 0 < t < 1, 2 < α ≤ 3, x(0) = x′(0) = 0, x′(1) = βx(ξ), where Dα 0+ denotes the standard Riemann-Liouville fractional derivative, 0 < ξ < 1 and 0 < β ξ < α− 1. Our analysis relies on the ...

In this paper, we devote to investigation of the existence of positive solutions for the boundary value problem of nonlinear fractional differential equations { D0+u(t)+ f (t,u(t)) = 0, 0 < t < 1, u(0) = u′(0) = · · ·u(n−2)(0) = D0+u(1), where D0+ , D β 0+ are the standard Riemann-Liouville fractional derivative, n− 1 < α n , n−2 β n−1 , n 3 . By means of constructing an exact cone of the Banac...

In this paper, we study the existence and nonexistence of positive solutions for the nonlinear differential equation of fractional order 0 ( ) ( ) ( ( )) 0, 0 1, 2 3, D u t a t f u t t α λ α + + = ≤ ≤ < ≤ where 0 D α + is the standard Riemann-Liouville fractional derivative, subject to the boundary conditions ( ) ( ) (0) (0) 0, (1) . u u u u u β η γ ξ ′ ′ ′ ′ = = = + Our analysis relies on Kras...

In this article, we establish certain sufficient conditions to show the existence of solutions a fractional differential equation with ?-Riemann-Liouville and ?-Caputo derivative in special Banach space. Our approach is based on fixed point theorems for Meir-Keeler condensing operators via measure non-compactness. Also an example given illustrate our approach.

In the finance market, it is well known that price change of underlying fractal transmission system can be modeled with Black-Scholes equation. This article deals finding approximate analytic solutions for time-fractional equation fractional integral boundary condition a European option pricing problem in Katugampola derivative sense. It generalizes both Riemann–Liouville and Hadamard derivativ...

This paper gathers the tools for solving Riemann-Liouville time fractional non-linear PDE’s by using a Galerkin method. method has advantage of not being more complicated than one used to solve same PDE with first order derivative. As model problem, existence and uniqueness is proved semilinear heat equations polynomial growth at infinity.

In this paper, using the Riemann-Liouville fractional q-integral, we establish some new fractional integral inequalities by using two parameters of deformation q1 and q2.