Mohammed Al-Refai

Maximum principle for the multi-term time-fractional diffusion equations with the Riemann-Liouville fractional derivatives

In this paper, the initial-boundary-value problems for linear and non-linear multi-term fractional diffusion equations with the Riemann-Liouville time-fractional derivatives are considered. To guarantee the uniqueness of solutions, we employ the weak and the strong maximum principles for the equations under consideration that are formulated and proved in this paper for the first time. An essential element of our proof of the maximum principles is an estimation for the value of the Riemann-Liouville fractional derivative of a function at its maximum point that is established in this paper for a wider space of functions compared to those used in our previous publications. In the linear case, the solutions to the problems under consideration are constructed in form of the Fourier series with respect to the eigenfunctions of the corresponding eigenvalue problems.

Maximum principle for the fractional diffusion equations with the Riemann-Liouville fractional derivative and its applications

In this paper, the initial-boundary-value problems for the one-dimensional linear and non-linear fractional diffusion equations with the Riemann-Liouville time-fractional derivative are analyzed. First, a weak and a strong maximum principles for solutions of the linear problems are derived. These principles are employed to show uniqueness of solutions of the initial-boundary-value problems for the non-linear fractional diffusion equations under some standard assumptions posed on the non-linear part of the equations. In the linear case and under some additional conditions, these solutions can be represented in form of the Fourier series with respect to the eigenfunctions of the corresponding Sturm-Liouville eigenvalue problems.

Traveling wave solutions using the variational method and the tanh method for nonlinear coupled equations

Abstract This paper applied both the tanh and the He’s variational iteration methods for analytic study for the nonlinear coupled Kortewge–de Vries (shortly, KdV) equations. Compared with existing sophisticated approaches, the proposed methods gives more general exact traveling wave solutions without much extra effort. Finally, we present an application to shallow water equations using the two methods, the calculations demonstrate the effectiveness and convenience of the He’s variational method for nonlinear coupled equations.

Fundamental Results of Conformable Sturm-Liouville Eigenvalue Problems

We suggest a regular fractional generalization of the well-known Sturm-Liouville eigenvalue problems. The suggested model consists of a fractional generalization of the Sturm-Liouville operator using conformable derivative and with natural boundary conditions on bounded domains. We establish fundamental results of the suggested model. We prove that the eigenvalues are real and simple and the eigenfunctions corresponding to distinct eigenvalues are orthogonal and we establish a fractional Rayleigh Quotient result that can be used to estimate the first eigenvalue. Despite the fact that the properties of the fractional Sturm-Liouville problem with conformable derivative are very similar to the ones with the classical derivative, we find that the fractional problem does not display an infinite number of eigenfunctions for arbitrary boundary conditions. This interesting result will lead to studying the problem of completeness of eigenfunctions for fractional systems.

An Efficient Series Solution for Fractional Differential Equations

We introduce a simple and efficient series solution for a class of nonlinear fractional differential equations of Caputos type. The new approach is a modified form of the well-known Taylor series expansion where we overcome the difficulty of computing iterated fractional derivatives, which do not compute in general. The terms of the series are determined sequentially with explicit formula, where only integer derivatives have to be computed. The efficiency of the new algorithm is illustrated through several examples. Comparison with other series methods such as the Adomian decomposition method and the homotopy perturbation method is made to indicate the efficiency of the new approach. The algorithm can be implemented for a wide class of fractional differential equations with different types of fractional derivatives.

Analysis of fractional diffusion equations of distributed order: Maximum principles and their applications

Abstract In this paper, we formulate and prove the weak and strong maximum principles for a general parabolic-type fractional differential operator with the Riemann–Liouville time-fractional derivative of distributed order. The proofs of the maximum principles are based on an estimate of the Riemann–Liouville fractional derivative at its maximum point that was recently derived by the authors. Some a priori norm estimates for solutions to initial-boundary value problems for linear and nonlinear fractional diffusion equations of distributed order and uniqueness results for these problems are presented.

Solving Fractional Diffusion Equation via the Collocation Method Based on Fractional Legendre Functions

A formulation of the fractional Legendre functions is constructed to solve the generalized time-fractional diffusion equation. The fractional derivative is described in the Caputo sense. The method is based on the collection Legendre and path following methods. Analysis for the presented method is given and numerical results are presented.

Theoretical and computational perspectives on the eigenvalues of fourth-order fractional Sturm–Liouville problem

ABSTRACT In this paper, we discuss a class of eigenvalue problems of fractional differential equations of order with variable coefficients. The method of solution is based on utilizing the fractional series solution to find theoretical eigenfunctions. Then, the eigenvalues are determined by applying the associated boundary conditions. A notable result, for certain cases, is that the eigenfunctions are characterized in terms of the Mittag-Leffler or semi Mittag-Leffler functions. The present findings demonstrate, for certain cases, the existence of a critical value at which the problem has no eigenvalue (for ), only one eigenvalue (at ), a finite or infinitely many eigenvalues (for ). The efficiency and accuracy of the present algorithm are demonstrated through several numerical examples.

Reduction of order formula and fundamental set of solutions for linear fractional differential equations

Abstract In this paper, we consider a class of linear fractional differential equations involving the Caputo–Fabrizio fractional derivative of order 1 α 2 . We derive, in closed form, a reduction of order formula to obtain a second linearly independent solution. We then establish a fundamental set of solutions result to the equation. One example is presented to illustrate the validity of the obtained results.

Monotonicity and convexity results for a function through its Caputo fractional derivative

Abstract In this paper we discuss certain geometric properties of a function through its Caputo fractional derivative. We show that convexity and monotonicity results can be obtained provided that the fractional derivative of the function is of one sign for some value of α. Analogous result for the global extrema of a function is obtained. However, to release the condition on the Diethelm paper [3] that the fractional derivative of the function is of one sign for all values of α in certain domain, higher order fractional inequalities are required. The applicability of the new results is discussed.