Fractional Two-point Boundary Value Problem Research Articles

New Series Approach Implementation for Solving Fuzzy Fractional Two-Point Boundary Value Problems Applications

In this work, the fuzzy fractional two-point boundary value problems (FFTBVPs) are analyzed and solved using the fuzzy fractional homotopy analysis method (FF-HAM). Fuzzy set theory mixed with Caputo fractional derivative properties is utilized to produce a new formulation of the standard HAM in the fuzzy domain for the persistence of approximation series solutions for fuzzy fractional differential equations with boundary conditions. The FF-HAM provides a suitable way of controlling the convergence of the series solution through the advantage of the convergence control parameter, which plays a pivotal role in solving a wide range of mathematical problems. The convergence analysis algorithm has been described, along with a graphical representation of the FF-HAM of the proposed applications. The method produces high accuracy solutions with simple implementation for solving linear and nonlinear fuzzy fractional boundary value problems associated with physical application. Also, the obtained results are analyzed and compared with those present in the literature to show the efficiency of the FF-HAM.

SCW-iterative-computational method for solving a wide class of nonlinear fractional optimal control problems with Caputo derivatives

The present paper offers an iterative-computational approach based on left and right operational matrices of the second Chebyshev wavelets (SCWs) in solving nonlinear fractional optimal control problems (FOCPs). The operational matrices are constructed by the hat functions (HFs). We first introduce necessary optimality conditions applying variational methods in a form of fractional two-point boundary value problem (TPBVP). For solving TPBVP, a fixed point iterative sequence, starting with an initial guess, in the hope of achieving more accurate solutions, is suggested. The SCWs-coefficients are easily obtained without any integrating. The order of convergence of the proposed method is accomplished in a new normed space. In the end of the article, some illustrative examples are provided to confirm the ability of the suggested technique.

The cubic B-spline interpolation method for numerical point solutions of conformable boundary value problems

In this analysis, the cubic B-spline method is employed for constructing the approximate solutions of a class of fractional two-point boundary value problems. These fractional problems are expressed in terms of the conformable fractional derivative approach. More deeply, a class of conformable Lane-Emden model is considering and modifying with singularities. Many numerical applications are presented and discussed to exhibit the feasibility and efficiency of the procedure involved in both linear and non-linear cases. The numerical findings are quite similar to those of the exact solutions as well as require relatively less computational work. Latterly, remarks and future research work are provided.

A Pseudo-Spectral Scheme for Systems of Two-Point Boundary Value Problems with Left and Right Sided Fractional Derivatives and Related Integral Equations

We target here to solve numerically a class of nonlinear fractional two-point boundary value problems involving left- and right-sided fractional derivatives. The main ingredient of the proposed method is to recast the problem into an equivalent system of weakly singular integral equations. Then, a Legendre-based spectral collocation method is developed for solving the transformed system. Therefore, we can make good use of the advantages of the Gauss quadrature rule. We present the construction and analysis of the collocation method. These results can be indirectly applied to solve fractional optimal control problems by considering the corresponding Euler–Lagrange equations. Two numerical examples are given to confirm the convergence analysis and robustness of the scheme.

An efficient collocation method with convergence rates based on Müntz spaces for solving nonlinear fractional two-point boundary value problems

The main purpose of this work is to develop effective spectral collocation methods for solving numerically general form of non-linear fractional two-point boundary value problems with two end-point singularities. Specifically, we define two classes of Muntz–Legendre polynomials to deal with the singular behaviors of the solutions at both end-points, which enhance greatly the accuracy of the numerical solutions. Important and practical formulas for ordinary derivatives of Muntz–Legendre polynomials are derived. Then using a three-term recurrence formula and Jacobi–Gauss quadrature rules, applicable and stable numerical approaches for computing left and right Caputo fractional derivatives of these basis functions are provided. One of the important features of the present method is to show how to recover spectral accuracy from the end-point singularities for the coupled systems of fractional differential equations with left and right fractional derivatives using the mentioned approximation basis. Moreover, we present a detailed analysis with accurate convergence rates of the singular behavior of the solution to a rather general system of nonlinear fractional differential equations including left and right fractional derivatives. Numerical results show the expected convergence rates. Finally, some numerical examples are given to demonstrate the performance of the proposed schemes.

Block Iterative Method for the Solution of Fractional Two-Point Boundary Value Problems

The goal of this paper deals with 4-Point Explicit Group (4-EG) iterative method as solver to solve the linear system which is generated by using Caputo’s finite difference approximation equation. Actually, this approximation equation was been generated from the discretization process of the fractional two-point boundary value problems via finite difference scheme and Caputo’s fractional operator. The formulation and implementation for 4-EG method have been also included. To investigate the efficiency of 4-EG method, this paper considered three examples numerical experiment. Based on numerical results, the finding showed that the 4-EG method is more efficient as compared with GS method in good agreement.

Solving state feedback control of fractional linear quadratic regulator systems using triangular functions

The present paper is useful for two reasons. First, as far as we know, the right operational matrix of the Riemann–Liouville fractional integral for triangular functions is introduced for the first time. So, hereafter, it can be applied in approximating any fractional optimal control problem in Caputo sense, over and over. Second, the rate of convergence of the triangular functions under a new-blown norm ‖.‖p,α in its corresponding space, denoted by Lp,α, is investigated. So far, the error analysis in Lp,α space has not been studied.In this article, both left and right operational matrices of the triangular functions for arbitrary fractional order integral α > 0 in Caputo sense are applied to approximate solutions of fractional linear optimal control systems which have a quadratic performance index. The necessary and sufficient optimality conditions are stated in a fractional two-point boundary value problem. This problem is converted to a set of coupled equations involving left and right Riemann–Liouville integral operators. Using these operational matrices of the triangular functions, an extension of the functions of control, state, co-state and other engaged functions is considered. This technique is a successful approach because of reducing such systems to a system of coupled Sylvester equations. Applying the definition of Kroneker product, a linear algebraic system containing the corresponding matrices of the original problem and the operational matrices of fractional order integral can be constructed. Accordingly, fractional linear quadratic optimal control can be solved indirectly. The advantage of this methodology, in addition to simplicity, is its low computational cost and its flexible precision. The simulation results confirm the reliability and validity of this method.

Caputo’s Finite Difference Solution of Fractional Two-Point BVPs Using AGE Iteration

The aim of this paper deals with the performance of Alternating Group Explicit (AGE) iterative method for solving the linear system based on Caputo’s finite difference approximation equation. Actually, the approximation equation was be generated from the discretization process of the fractional two-point boundary value problems by using finite difference discretization scheme and Caputo’s operator. The formulation and implementation of AGE iterative method have been included. To investigate the efficiency of AGE iterative method, this paper considered four examples of numerical experiments. Based on numerical results, the findings showed that the AGE method is more efficient as compared with GS method.

A Hybrid Spectral Element Method for Fractional Two-Point Boundary Value Problems

AbstractWe propose a hybrid spectral element method for fractional two-point boundary value problem (FBVPs) involving both Caputo and Riemann-Liouville (RL) fractional derivatives. We first formulate these FBVPs as a second kind Volterra integral equation (VIEs) with weakly singular kernel, following a similar procedure in [16]. We then design a hybrid spectral element method with generalized Jacobi functions and Legendre polynomials as basis functions. The use of generalized Jacobi functions allow us to deal with the usual singularity of solutions att= 0. We establish the existence and uniqueness of the numerical solution, and derive a hptype error estimates underL2(I)-norm for the transformed VIEs. Numerical results are provided to show the effectiveness of the proposed methods.

New Modified Adomian Decomposition Recursion Schemes for Solving Certain Types of Nonlinear Fractional Two-Point Boundary Value Problems

We apply new modified recursion schemes obtained by the Adomian decomposition method (ADM) to analytically solve specific types of two-point boundary value problems for nonlinear fractional order ordinary and partial differential equations. The new modified recursion schemes, which sometimes utilize the technique of Duan’s convergence parameter, are derived using the Duan-Rach modified ADM. The Duan-Rach modified ADM employs all of the given boundary conditions to compute the remaining unknown constants of integration, which are then embedded in the integral solution form before constructing recursion schemes for the solution components. New modified recursion schemes obtained by the method are generated in order to analytically solve nonlinear fractional order boundary value problems with a variety of two-point boundary conditions such as Robin and separated boundary conditions. Some numerical examples of such problems are demonstrated graphically. In addition, the maximal errors (MEn) or the error remainder functions (ERn(x)) of each problem are calculated.

A Neural Network Approach for Solving a Class of Fractional Optimal Control Problems

In this paper the perceptron neural networks are applied to approximate the solution of fractional optimal control problems. The necessary (and also sufficient in most cases) optimality conditions are stated in a form of fractional two-point boundary value problem. Then this problem is converted to a Volterra integral equation. By using perceptron neural network’s ability in approximating a nonlinear function, first we propose approximating functions to estimate control, state and co-state functions which they satisfy the initial or boundary conditions. The approximating functions contain neural network with unknown weights. Using an optimization approach, the weights are adjusted such that the approximating functions satisfy the optimality conditions of fractional optimal control problem. Numerical results illustrate the advantages of the method.

Existence and uniqueness of positive solutions for boundary value problems of a fractional differential equation with a parameter

In this paper, we are concerned with the existence and uniqueness of positive solutions for the following nonlinear fractional two-point boundary value problem  D α 0+u(t) + λf(t, u(t), u(t)) = 0, 0 < t < 1, 2 < α ≤ 3, u(0) = u 0 (0) = u 0 (1) = 0, where D α 0+ is the standard Riemann-Liouville fractional derivative, and λ is a positive parameter. Our analysis relies on a fixed point theorem and some properties of eigenvalue problems for a class of general mixed monotone operators. Our results can not only guarantee the existence of a unique positive solution, but also be applied to construct an iterative scheme for approximating it. An example is given to illustrate the main results.

A Bessel collocation method for solving fractional optimal control problems

In the present paper, we apply the truncated Bessel series approximation by using collocation scheme, for solving linear and nonlinear fractional optimal control problems (OCPs) indirectly. Therefore, the necessary (and also sufficient in most cases) optimality conditions are stated in a form of nonlinear (or linear) fractional two-point boundary value problem (TPBVP). For solving this mentioned TPBVP, we generalize a new numerical method (which is called the Bessel collocation method). One of the best advantages of this generalization is that, there is no need to use operational matrices of differentiation and also the new generalized idea can be implemented in any mathematical software. Some numerical examples are provided to confirm the accuracy of the proposed method. All of the numerical computations have been performed on a PC using several programs written in MAPLE 13.

Solvability of a Class of Higher-Order Fractional Two-Point Boundary Value Problem

This paper investigates the solvability of a class of higher-order fractional two-point boundary value problem (BVP), and presents several new results. First, Green’s function of the considered BVP is obtained by using the property of Caputo derivative. Second, based on Schaefer’s fixed point theorem, the solvability of the considered BVP is studied, and a sufficient condition is presented for the existence of at least one solution. Finally, an illustrative example is given to support the obtained new results.

Solving fractional two-point boundary value problems using continuous analytic method

In this article, the homotopy analysis method is applied to provide approximate solutions for linear and nonlinear two-point boundary value problems of fractional order. The solution was calculated in the form of a convergent power series with easily computable components. In this method, one has great freedom to select auxiliary functions, operators, and parameters in order to ensure the convergence of the approximate solution and to increase both the rate and region of convergence. Numerical examples are provided to demonstrate the accuracy and efficiency of the present method. Meanwhile, further iterations can produce more accurate results and decrease the error.

Uniqueness of Positive Solutions for a Perturbed Fractional Differential Equation

We are concerned with the existence and uniqueness of positive solutions for the following nonlinear perturbed fractional two-point boundary value problem:D0+αu(t)+f(t,u,u',…,u(n-2))+g(t)=0, 0&lt;t&lt;1, n-1&lt;α≤n, n≥2,u(0)=u'(0)=⋯=u(n-2)(0)=u(n-2)(1)=0, whereD0+αis the standard Riemann-Liouville fractional derivative. Our analysis relies on a fixed-point theorem of generalized concave operators. An example is given to illustrate the main result.