Numerical Solution of the Distributed-Order Fractional Bagley-Torvik Equation

In this paper, two numerical methods are proposed for solving distributed-order fractional Bagley-Torvik equation.This equation is used in modeling the motion of a rigid plate immersed in a Newtonian fluid with respect to the nonnegative density function. Using the composite Boole's rule the distributedorder Bagley-Torvik equation is approximated by a multi-term time-fractional equation, which is then solved by the GrunwaldLetnikov method(GLM) and the fractional differential transform method(FDTM). Finally, we compared our results with the exact results of some cases and show the excellent agreement between the approximate result and the exact solution.

Hermite Solution of Bagley-Torvik Equation of Fractional Order

In this paper, a new methodology of fractional derivatives based upon Hermite polynomial is projected. The fractional derivatives are demonstrated according to Caputo sense. Hermite collocation technique is introduced to express the definite results of Bagley-Torvik Equations. The appropriateness and straightforwardness of numerical plan is presented by graphs and error tables.

Exact Solution to Nonlinear Differential Equations of Fractional Order via (<i>G’</i>/<i>G</i>)-Expansion Method

In this article, a new application to find the exact solutions of nonlinear partial time-space fractional differential Equation has been discussed. Firstly, the fractional complex transformation has been implemented to convert nonlinear partial fractional differential Equations into nonlinear ordinary differential Equations. Afterwards, the (G'/G)-expansion method has been implemented, to celebrate the exact solutions of these Equations, in the sense of modified Riemann-Liouville derivative. As application, the exact solutions of time-space fractional Burgers’ Equation have been discussed.

A Novel Analytical Technique of the Fractional Bagley-Torvik Equations for Motion of a Rigid Plate in Newtonian Fluids

The current paper is concerned with a modified Homotopy perturbation technique.This modification allows achieving an exact solution of an initial value problem of the fractional differential equation.The approach is powerful,effective,and promising in analyzing some classes of fractional differential equations for heat conduction problems and other dynamical systems.To crystallize the new approach,some illustrated examples are introduced.

EXISTENCE AND UNIQUENESS RESULTS FOR BOUNDARY VALUE PROBLEMS OF HIGHER ORDER FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS INVOLVING GRONWALL'S INEQUALITY IN BANACH SPACES

We study boundary value problems for fractional integro-differential equations involving Caputo derivative of order α∈ （n-1, n） in Banach spaces. Existence and uniqueness results of solutions are established by virtue of the Holder＇s inequality, a suitable singular Cronwall＇s inequality and fixed point theorem via a priori estimate method. At last, examples are given to illustrate the results.

Existence and Uniqueness of Solution for a Fractional Order Integro-Differential Equation with Non-Local and Global Boundary Conditions

In this paper, we prove an important existence and uniqueness theorem for a fractional order Fredholm – Volterra integro-differential equation with non-local and global boundary conditions by converting it to the corresponding well known Fredholm integral equation of second kind. The considered in this paper has been solved already numerically in [1].

Solving the Nonlinear Variable Order Fractional Differential Equations by Using Euler Wavelets

An Euler wavelets method is proposed to solve a class of nonlinear variable order fractional differential equations in this paper.The properties of Euler wavelets and their operational matrix together with a family of piecewise functions are first presented.Then they are utilized to reduce the problem to the solution of a nonlinear system of algebraic equations.And the convergence of the Euler wavelets basis is given.The method is computationally attractive and some numerical examples are provided to illustrate its high accuracy.

A New Approach for the Exact Solutions of Nonlinear Equations of Fractional Order via Modified Simple Equation Method

In this article, the modified simple equation method has been extended to celebrate the exact solutions of nonlinear partial time-space differential equations of fractional order. Firstly, the fractional complex transformation has been implemented to convert nonlinear partial fractional differential equations into nonlinear ordinary differential equations. Afterwards, modified simple equation method has been implemented, to find the exact solutions of these equations, in the sense of modified Riemann-Liouville derivative. For applications, the exact solutions of time-space fractional derivative Burgers’ equation and time-space fractional derivative foam drainage equation have been discussed. Moreover, it can also be concluded that the proposed method is easy, direct and concise as compared to other existing methods.

A Galerkin-Type Fractional Approach for Solutions of Bagley-Torvik Equations

In this study,we present a numerical scheme similar to the Galerkin method in order to obtain numerical solutions of the Bagley Torvik equation of fractional order 3/2.In this approach,the approximate solution is assumed to have the form of a polynomial in the variable t=xα,whereαis a positive real parameter of our choice.The problem is firstly expressed in vectoral form via substituting the matrix counterparts of the terms present in the equation.After taking inner product of this vector with nonnegative integer powers of t up to a selected positive parameter N,a set of linear algebraic equations is obtained.After incorporation of the boundary conditions,the approximate solution of the problem is then computed from the solution of this linear system.The present method is illustrated with two examples.

NUMERICAL SIMULATIONS FOR A VARIABLE ORDER FRACTIONAL CABLE EQUATION

In this article, Crank-Nicolson method is used to study the variable order fractional cable equation. The variable order fractional derivatives are described in the Riemann- Liouville and the Griinwald-Letnikov sense. The stability analysis of the proposed technique is discussed. Numerical results are provided and compared with exact solutions to show the accuracy of the proposed technique.

SOLUTIONS OF THE BENJAMIN-ONO EQUATION IN FRACTIONAL ORDER SOBOLEV SPACES

We show for the Benjamin-Ono equation an existence uniqeness theorem in Sobolev spaces of arbitrary fractional order s greater-than-or-equal-to 2, provided the initial data is given in the same space.

Higher-order approximate solutions of fractional stochastic point kinetics equations in nuclear reactor dynamics

Stochastic point kinetics equations(SPKEs) are a system of Ito? stochastic differential equations whose solution has been obtained by higher-order approximation.In this study, a fractional model of SPKEs has been analyzed. The efficiency of the proposed higher-order approximation scheme has been discussed in the results section. The solutions of SPKEs in the presence of Newtonian temperature feedback have also been provided to further discuss the physical behavior of the fractional model.

Solving a Nonlinear Multi-Order Fractional Differential Equation Using Legendre Pseudo-Spectral Method

In this paper, we apply the Legendre spectral-collocation method to obtain approximate solutions of nonlinear multi-order fractional differential equations (M-FDEs). The fractional derivative is described in the Caputo sense. The study is conducted through illustrative example to demonstrate the validity and applicability of the presented method. The results reveal that the proposed method is very effective and simple. Moreover, only a small number of shifted Legendre polynomials are needed to obtain a satisfactory result.

Reduced-order finite element method based on POD for fractional Tricomi-type equation

The reduced-order finite element method （FEM） based on a proper orthogo- nal decomposition （POD） theory is applied to the time fractional Tricomi-type equation. The present method is an improvement on the general FEM. It can significantly save mem- ory space and effectively relieve the computing load due to its reconstruction of POD basis functions. Furthermore, the reduced-order finite element （FE） scheme is shown to be un- conditionally stable, and error estimation is derived in detail. Two numerical examples are presented to show the feasibility and effectiveness of the method for time fractional differential equations （FDEs）.

Multiple Solutions for a Class of Variable-Order Fractional Laplacian Equations with Concave-Convex Nonlinearity

This paper is concerned with the following variable-order fractional Laplacian equations , where N ≥ 1 and N > 2s(x,y) for (x,y) ∈ Ω × Ω, Ω is a bounded domain in RN, s(⋅) ∈ C (RN × RN, (0,1)), (-Δ)s(⋅) is the variable-order fractional Laplacian operator, λ, μ > 0 are two parameters, V: Ω → [0, ∞) is a continuous function, f ∈ C(Ω × R) and q ∈ C(Ω). Under some suitable conditions on f, we obtain two solutions for this problem by employing the mountain pass theorem and Ekeland’s variational principle. Our result generalizes the related ones in the literature.

CONVERGENCE ANALYSIS OF THE JACOBI SPECTRAL-COLLOCATION METHOD FOR FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS

We propose and analyze a spectral Jacobi-collocation approximation for fractional order integro-differential equations of Volterra type. The fractional derivative is described in the Caputo sense. We provide a rigorous error analysis for the collection method, which shows that the errors of the approximate solution decay exponentially in L^∞ norm and weighted L^2-norm. The numerical examples are given to illustrate the theoretical results.

Existence of positive solutions for integral boundary value problem of fractional differential equations

In this paper,we concern ourselves with the existence of positive solutions for a type of integral boundary value problem of fractional differential equations with the fractional order linear derivative operator. By using the fixed point theorem in cone,the existence of positive solutions for the boundary value problem is obtained. Some examples are also presented to illustrate the application of our main results.

Exact solutions of nonlinear fractional differential equations by (G'/G)-expansion method

In this paper, we use the fractional complex transform and the （G＇/G）-expansion method to study the nonlinear fractional differential equations and find the exact solutions. The fractional complex transform is proposed to convert a partial fractional differential equation with Jumarie＇s modified Riemann-Liouville derivative into its ordinary differential equation. It is shown that the considered transform and method are very efficient and powerful in solving wide classes of nonlinear fractional order equations.

Enriched reproducing kernel particle method for fractional advection–diffusion equation

The reproducing kernel particle method （RKPM） has been efficiently applied to problems with large deformations, high gradients and high modal density. In this paper, it is extended to solve a nonlocal problem modeled by a fractional advectiondiffusion equation （FADE）, which exhibits a boundary layer with low regularity. We formulate this method on a moving least-square approach. Via the enrichment of fractional-order power functions to the traditional integer-order basis for RKPM, leading terms of the solution to the FADE can be exactly reproduced, which guarantees a good approximation to the boundary layer. Numerical tests are performed to verify the proposed approach.

Existence and Uniqueness for the Boundary Value Problems of Nonlinear Fractional Differential Equation

This paper studies the existence and uniqueness of solutions for a class of boundary value problems of nonlinear fractional order differential equations involving the Caputo fractional derivative by employing the Banach’s contraction principle and the Schauder’s fixed point theorem. In addition, an example is given to demonstrate the application of our main results.