Boundary Value Problems Of Fractional Order Research Articles

On the rate of convergence of spectral collocation methods for nonlinear multi-order fractional initial value problems

Multi-order fractional differential equations are motivated by their flexibility to describe complex multi-rate physical processes. This paper is concerned with the convergence behavior of a spectral collocation method when used to approximate solutions of nonlinear multi-order fractional initial value problems. The collocation scheme and its convergence analysis are developed based on the novel spectral collocation method which has been recently presented by Wang et al. (J Sci Comput 76(1):166–188, 2018) for single fractional-order boundary value problems. The proposed method is an indirect approach since we act on the equivalent Volterra integral equation of the second kind. More precisely, the spectral rate of convergence for the proposed method is established in the $$L^2 $$- and $$ L^{\infty } $$-norms. The method enjoys high accuracy for problems with smooth solutions. Exponentially rapid convergence is observed with a small number of degree of freedoms and for all samples of fractional orders. Numerical examples are presented to support the theoretical finding.

A Novel Method for Solutions of Fourth-Order Fractional Boundary Value Problems

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.

Positive solutions for an infinite system of fractional order boundary value problems

In this paper, we investigate the existence of positive solutions for an infinite system of fractional order boundary value problems in a Banach sequence space c. Our analysis relies on the Krasnosel’skii fixed point theorem in a cone in conjunction with the criterion of relative compactness in C^{1}(J, mathbf{c}). Finally, an example is given to illustrate our abstract results.

Dhage iterative principle for quadratic perturbation of fractional boundary value problems with finite delay

In this article, by employing Dhage iterative method embodied in current hybrid fixed point theorem (HFPT) of Dhage, we derive an algorithm for the numerical solutions via construction of a sequence of successive approximations for a fractional order boundary value problem (FBVP) with finite delay. By using this technique, we obtain existence as well as approximation of solutions under weaker partial Lipschitz and partial compactness type conditions in a partially ordered Banach space. Additionally, we prove an existence and uniqueness theorem under a weaker partial nonlinear Lipschitz condition. The assumptions and main outcomes are also illustrated by two examples.

Existence and Stability Results for a Fractional Order Differential Equation with Non-Conjugate Riemann-Stieltjes Integro-Multipoint Boundary Conditions

We discuss the existence and uniqueness of solutions for a Caputo-type fractional order boundary value problem equipped with non-conjugate Riemann-Stieltjes integro-multipoint boundary conditions on an arbitrary domain. Modern tools of functional analysis are applied to obtain the main results. Examples are constructed for the illustration of the derived results. We also investigate different kinds of Ulam stability, such as Ulam-Hyers stability, generalized Ulam-Hyers stability, and Ulam-Hyers-Rassias stability for the problem at hand.

Existence of solutions for a higher order fractional boundary value problem posed on the half-line

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.

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

In this article, we have explored the numerical solution of fourth order fractional boundary value problems, involving product terms, by means of quintic spline collocation method. The proposed numerical approach is based on non-polynomial quintic spline functions comprised of a trigonometric part and polynomial part. The second and fourth order convergence of the presented algorithm has been discussed rigorously. Some test examples have been considered and the approximate results are found to be more accurate as compared to the other variants on the topic.

Strongly positive-like operators and eigenvalue criteria for existence and nonexistence of positive solutions for $\alpha$-order fractional boundary value problems

Strongly positive-like operators and eigenvalue criteria for existence and nonexistence of positive solutions for $\alpha$-order fractional boundary value problems

Legendre wavelet operational matrix method for solving fractional differential equations in some special conditions

This paper proposes a new technique which rests upon Legendre wavelets for solving linear and non-linear forms of fractional order initial and boundary value problems. In some particular circumstances, a new operational matrix of fractional derivative is generated by utilizing some significant properties of wavelets and orthogonal polynomials. We approached the solution in a finite series with respect to Legendre wavelets and then by using these operational matrices, we reduced the fractional differential equations into a system of algebraic equations. Finally, the introduced technique is tested on several illustrative examples. The obtained results demonstrate that this technique is a very impressive and applicable mathematical tool for solving fractional differential equations.

Existence results for positive solutions to iterative systems of four-point fractional-order boundary value problems in a Banach space

We investigate the eigenvalue intervals of [Formula: see text] for which the iterative system of four-point fractional-order boundary value problem has at least one positive solution by utilizing Guo–Krasnosel’skii fixed point theorem under suitable conditions. The obtained results in the paper are illustrated with an example for their feasibility.

Ulam stability to a toppled systems of nonlinear implicit fractional order boundary value problem

In this manuscript, we give some sufficient conditions for existence, uniqueness and various kinds of Ulam stability for a toppled system of fractional order boundary value problems involving the Riemann–Liouville fractional derivative. Applying the Banach contraction principle and the Leray–Schauder result of cone type, uniqueness and existence results are proved for the proposed toppled system. Stability is investigated by using the classical technique of nonlinear functional analysis. The results obtained are well illustrated with the aid of an example.

Existence results for fractional order differential equation with nonlocal Erd\xe9lyi\u2013Kober and generalized Riemann\u2013Liouville type integral boundary conditions at resonance

In this paper, we discuss a nonlinear fractional order boundary value problem with nonlocal Erdélyi–Kober and generalized Riemann–Liouville type integral boundary conditions. By using Mawhin continuation theorem, we investigate the existence of solutions of this boundary value problem at resonance. It is shown that the boundary value problem \t\t\tDqcx(t)=f(t,x(t),x′(t)),t∈[0,T],1<q≤2,x(0)=αIηγ,δx(ζ),x(T)=βρIpx(ξ),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{gathered} {}^{c}D^{q}x(t)=f \\bigl(t, x(t),x'(t) \\bigr),\\quad t \\in[0,T], 1< q\\leq2, \\\\ x(0)=\\alpha I^{\\gamma,\\delta}_{\\eta}x(\\zeta),\\qquad x(T)=\\beta{}^{\\rho }I^{p}x( \\xi),\\end{gathered} $$\\end{document} has at least one solution under some suitable conditions, where alpha, betainmathbb{R}, 0<zeta, xi<T.

Ulam stability results for the solutions of nonlinear implicit fractional order differential equations

In this manuscript, we study the existence and uniqueness of solution for a class of fractional order boundary value problem (FBVP) for implicit fractional differential equations with Riemann-Liouville derivative. Furthermore, we investigate different kinds of Ulam stability such as Ulam-Hyers stability, generalized Ulam-Hyers stability, Ulam-Hyers-Rassias stability and generalized Ulam-Hyers-Rassias stability for the proposed problem. The concerned analysis is carried out through using classical technique of nonlinear functional analysis. The main results are illustrated by providing a couple of examples

Existence theory for fractional differential equations with non-separated type nonlocal multi-point and multi-strip boundary conditions

We introduce a more general class of fractional-order boundary value problems involving non-separated type multi-point and multi-strip boundary conditions. Several existence and uniqueness results for the given problem are established by applying the tools of fixed-point theory. Some illustrative examples are also included. The boundary conditions introduced in this work are of quite general nature and reduce to many special cases by fixing the parameters involved in the conditions.

Fractional-order boundary value problems with Katugampola fractional integral conditions

In this paper, we study existence (uniqueness) of solutions for nonlinear fractional differential equations with Katugampola fractional integral conditions. Several fixed point theorems are used for sufficient conditions of existence (uniqueness) solutions of nonlinear differential equations such as Banach’s contraction principle, the Leray–Schauder nonlinear alternative, and Krasnoselskii’s fixed point theorem. Applications of the main results are also presented.

On the Theory of Fractional Calculus in the Pettis-Function Spaces

Throughout this paper, we outline some aspects of fractional calculus in Banach spaces. Some examples are demonstrated. In our investigations, the integrals and the derivatives are understood as Pettis integrals and the corresponding derivatives. Our results here extended all previous contributions in this context and therefore are new. To encompass the full scope of our paper, we show that a weakly continuous solution of a fractional order integral equation, which is modeled off some fractional order boundary value problem (where the derivatives are taken in the usual definition of the Caputo fractional weak derivative), may not solve the problem.

Multiple positive solutions of fractional-order boundary value problem with integral boundary conditions

Multiple positive solutions of fractional-order boundary value problem with integral boundary conditions

On a Fractional Oscillator Equation with Natural Boundary Conditions

We prove existence of solutions for a nonlinear fractional oscillator equation with both left Riemann-Liouville and right Caputo fractional derivatives subject to natural boundary conditions. The proof is based on a transformation of the problem into an equivalent lower order fractional boundary value problem followed by the use of an upper and lower solutions method. To succeed with such approach, we first prove a result on the monotonicity of the right Caputo derivative.

Nonlocal semi-linear fractional-order boundary value problems with strip conditions

This paper is concerned with the question of existence of solutions for one-dimensional higher-order semi-linear fractional differential equations supplemented with nonlocal strip type boundary conditions. The nonlocal strip condition addresses a situation where the linear combination of the values of unknown function at two nonlocal points, located to the left and right hand sides of the strip, respectively, is proportional to its strip value. The case of Stieltjes type strip condition is also discussed. Our results, relying on some standard fixed point theorems are supported with illustrative examples.

Higher order fractional boundary value problems for mixed type derivatives

Higher order fractional boundary value problems for mixed type derivatives