Solution of Third Order Fractional Boundary Value Problems by Using Greens Function Method

A new reproducing kernel method for variable order fractional boundary value problems for functional differential equations

The aim of this paper is to study the existence and uniqueness of solutions for a boundary value problem associated with a fractional nonlinear differential equation with higher order posed on the half-line. An appropriate continuous embedding for suitable Banach spaces are proved and the Minty–Browder theorem for monotone operators is used in the proof of existence of solutions for a boundary value problem of fractional order posed on the half-line.

PurposeThe purpose of this paper is to find approximate solutions for a general class of fractional order boundary value problems that arise in engineering applications.Design/methodology/approachA newly developed semi-analytical scheme will be applied to find approximate solutions for fractional order boundary value problems. The technique is regarded as an extension of the well-established variation iteration method, which was originally proposed for initial value problems, to cover a class of boundary value problems.FindingsIt has been demonstrated that the method yields approximations that are extremely accurate and have uniform distributions of error throughout their domain. The numerical examples confirm the method’s validity and relatively fast convergence.Originality/valueThe generalized variational iteration method that is presented in this study is a novel strategy that can handle fractional boundary value problem more effectively than the classical variational iteration method, which was designed for initial value problems.

Abstract\n The existence criterion has been extensively studied for different classes in fractional differential equations (FDEs) through different mathematical methods. The class of fractional order boundary value problems (FOBVPs) with p-Laplacian operator is one of the most popular class of the FDEs which have been recently considered by many scientists as regards the existence and uniqueness. In this scientific work our focus is on the existence and uniqueness of the FOBVP with p-Laplacian operator of the form: \n \n \n \n \n \n \n D\n γ\n \n (\n \n ϕ\n p\n \n (\n \n D\n θ\n \n z\n (\n t\n )\n )\n )\n +\n a\n (\n t\n )\n f\n (\n z\n (\n t\n )\n )\n =\n 0\n \n \n $D^{\\gamma}(\\phi_{p}(D^{\\theta}z(t)))+a(t)f(z(t)) =0$\n , \n \n \n \n \n \n 3\n <\n θ\n \n \n $3<{\\theta}$\n , \n \n \n \n \n \n γ\n ≤\n 4\n \n \n $\\gamma\\leq{4}$\n , \n \n \n \n \n \n t\n ∈\n [\n 0\n ,\n 1\n ]\n \n \n $t\\in[0,1]$\n , \n \n \n \n \n \n z\n (\n 0\n )\n =\n \n z\n ‴\n \n (\n 0\n )\n \n \n $z(0)=z'''(0)$\n , \n \n \n \n \n \n η\n \n D\n α\n \n z\n (\n t\n )\n \n |\n \n t\n =\n 1\n \n \n =\n \n z\n ′\n \n (\n 0\n )\n \n \n $\\eta D^{\\alpha}z(t)|_{t=1}= z'(0)$\n , \n \n \n \n \n \n ξ\n \n z\n ″\n \n (\n 1\n )\n −\n \n z\n ″\n \n (\n 0\n )\n =\n 0\n \n \n $\\xi z''(1)-z''(0)=0$\n , \n \n \n \n \n \n 0\n <\n α\n <\n 1\n \n \n $0<\\alpha<1$\n , \n \n \n \n \n \n \n ϕ\n p\n \n (\n \n D\n θ\n \n z\n (\n t\n )\n )\n \n |\n \n t\n =\n 0\n \n \n =\n 0\n =\n \n \n (\n \n ϕ\n p\n \n (\n \n D\n θ\n \n z\n (\n t\n )\n )\n )\n \n ′\n \n \n |\n \n t\n =\n 0\n \n \n \n \n $\\phi_{p}(D^{\\theta}z(t))|_{t=0}=0 =(\\phi_{p}(D^{\\theta}z(t)))'|_{t=0}$\n , \n \n \n \n \n \n \n \n (\n \n ϕ\n p\n \n (\n \n D\n θ\n \n z\n (\n t\n )\n )\n )\n \n ″\n \n \n |\n \n t\n =\n 1\n \n \n =\n \n 1\n 2\n \n \n \n (\n \n ϕ\n p\n \n (\n \n D\n θ\n \n z\n (\n t\n )\n )\n )\n \n ″\n \n \n |\n \n t\n =\n 0\n \n \n \n \n $(\\phi_{p}(D^{\\theta} z(t)))''|_{t=1} = \\frac{1}{2}(\\phi_{p}(D^{\\theta} z(t)))''|_{t=0}$\n , \n \n \n \n \n \n \n \n (\n \n ϕ\n p\n \n (\n \n D\n θ\n \n z\n (\n t\n )\n )\n )\n \n ‴\n \n \n |\n \n t\n =\n 0\n \n \n =\n 0\n \n \n $(\\phi_{p}(D^{\\theta}z(t)))'''|_{t=0}=0$\n , where \n \n \n \n \n \n 0\n <\n ξ\n ,\n η\n

Fractional order boundary value problem with integral boundary conditions involving Pettis integral

In this paper, by using the coincidence degree theory, we consider the following boundary value problem for fractional differential equation where denotes the Caputo fractional differential operator of order α, 2 < α ≤ 3. A new result on the existence of solutions for above fractional boundary value problem is obtained. Mathematics Subject Classification (2000): 34A08, 34B15.

In this paper, we find the solutions of fourth order fractional boundary value problems by using the reproducing kernel Hilbert space method. Firstly, the reproducing kernel Hilbert space method is introduced and then the method is applied to this kind problems. The experiments are discussed and the approximate solutions are obtained to be more correct compared to the other obtained results in the literature.

In this paper, a sinc-collocation method is described to determine the approximate solution of fractional order boundary value problem (FBVP). The results obtained are presented as two new theorems. The fractional derivatives are defined in the Caputo sense, which is often used in fractional calculus. In order to demonstrate the efficiency and capacity of the present method, it is applied to some FBVP with variable coefficients. Obtained results are compared to exact solutions as well as Cubic Spline solutions. The comparisons can be used to conclude that sinc-collocation method is powerful and promising method for determining the approximate solutions of FBVPs in different types of scenarios.

In this article, we discuss the existence of solutions of a fractional boundary value problem of order m ∈ (1, 2], with nonlocal non-separated type integral multipoint boundary conditions. Shaefer type and Krasnoselskii’s fixed point theorems are used to prove existence results for the given problem. To establish the uniqueness of solutions Banach contraction principle is used. The criteria for HyersUlam stability of the given boundary value problem is also discussed. Some examples are included for the illustration of our results.

We consider a higher order fractional boundary value problem with Lidstone like boundary conditions, where the nonlinearity is an L1-Carathèodory function. We first consider the lower order problem. Then, by using a convolution to construct the Green’s function for the higher order problem, we are able to apply a recent fixed point theorem to show the existence of positive solutions of the boundary value problem.

This paper consider the existence of at least one positive solution of a Riemann-Liouville fractional delay singular boundary value problem with sign-changing nonlinerty. To establish sufficient conditions we use the Guo-Krasnosel?skii fixed point theorem.

In this article, we generalize the Legendre wavelets operational matrix of derivatives to fractional order derivatives in the Caputo sense. Legendre wavelets and their properties are employed for deriving Legendre wavelets operational matrix of fractional derivatives and a general procedure for forming this matrix is introduced. Then truncated Legendre wavelets expansions together with these matrices are used for numerical solution of Bagley–Torvik fractional order boundary value problems. Several examples are included to demonstrate accuracy and applicability of the proposed method. Key words: Shifted Legendre polynomials, Legendre wavelets, Caputo derivative, fractional order boundary value problems.

In this paper we present an approximate solution of a fractional order two-point boundary value problem (FBVP). We use the sinc-Galerkin method that has almost not been employed for the fractional order differential equations. We expand the solution function in a finite series in terms of composite translated sinc functions and some unknown coefficients. These coefficients are determined by writing the original FBVP as a bilinear form with respect to some base functions. The bilinear forms are expressed by some appropriate integrals. These integrals are approximately solved by sinc quadrature rule where a conformal map and its inverse are evaluated at sinc grid points. Obtained results are presented as two new theorems. In order to illustrate the applicability and accuracy of the present method, the method is applied to some specific examples, and simulations of the approximate solutions are provided. The results are compared with the ones obtained by the Cubic splines. Because there are only a few studies regarding the application of sinc-type methods to fractional order differential equations, this study is going to be a totally new contribution and highly useful for the researchers in fractional calculus area of scientific research.

Non-polynomial quintic spline for solving fourth-order fractional boundary value problems involving product terms

Positive solutions for a fractional boundary value problem