Existence and uniqueness of solutions for fractional order m-point boundary value problems

Shah, Kamal; Zeb, Salman; Khan, Rahmat Ali

doi:10.7153/fdc-05-15

Cited by 11 publications

(7 citation statements)

References 10 publications

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“…Beside this we also provide some results from fixed point theory to transfer our model into fixed point problem. For beginner of the field we provide some useful books, articles therefore the interested readers are referred to see [8,9,12,15,24]. Definition 2.1.…”

Section: Basic Results and Fundamental Definitionsmentioning

confidence: 99%

Application of topological degree method in quantitative behavior of fractional differential equations

Ghaus

Ahmad

Haq

2020

Filomat

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In the present manuscript we incorporate fractional order Caputo derivative to study a class of non-integer order differential equation. For existence and uniqueness of solution some results from fixed point theory is on our disposal. The method used for exploring these existence results is topological degree method and some auxiliary conditions are developed for stability analysis. For further elaboration an illustrative example is provided in the last part of the research article.

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Section: Basic Results and Fundamental Definitionsmentioning

confidence: 99%

Application of topological degree method in quantitative behavior of fractional differential equations

Ghaus

Ahmad

Haq

2020

Filomat

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“…The qualitative theory devoted to existence of solutions to non-integer order differential equations involving boundary conditions has been an active area of research for the last few decades. By using various tools and method of functional analysis and fixed point theory, the concerned theory has been explored very well, for detail see [1,2,22,23]. However, in the mentioned papers, the concerned conditions for existence of solutions required compactness of the operator which restrict the area to some limited classes of fractional differential equations.…”

Section: Introductionmentioning

confidence: 99%

Existence results and Hyers-Ulam stability to a class of nonlinear arbitrary order differential equations

Ali¹,

Shah²,

Khan³

2017

J. Nonlinear Sci. Appl.

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This paper is concerned with developing some conditions that reveal existing and stability analysis for solutions to a class of differential equations with fractional order. The required conditions are obtained by applying the technique of degree theory of topological type. The concerned problem is converted to the integral equation and then to operator equation, where the operator is defined by T : C[0, 1] → C[0, 1]. It should be noted that the assumptions on nonlinear function f(t, u(t)) does not usually ascertain that the operator T being compact. Moreover, in this paper we also establish some conditions under which the solution of the considered class is Hyers-Ulam stable and also satisfies the conditions of Hyers-Ulam-Rassias and generalized Hyers-Ulam stability. Proper example is provided for the illustration of main results.

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“…The analytical results based on the existence and uniqueness of solutions to some fractional differential equations have been investigated by many authors (see, for example, [25,45,[50][51][52] and the references therein). Bearing in mind the increasing application of fractional-order differential equations and PDEs, and due to the computational complexities of fractional calculus and the non-availability of their explicit analytic solutions, the need to exploit various efficient and reliable numerical schemes is a problem of fundamental interest.…”

Section: Introductionmentioning

confidence: 99%

A generalized scheme based on shifted Jacobi polynomials for numerical simulation of coupled systems of multi-term fractional-order partial differential equations

Shah¹,

Khalil²,

Khan³

2017

LMS J. Comput. Math.

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Due to the increasing application of fractional calculus in engineering and biomedical processes, we analyze a new method for the numerical simulation of a large class of coupled systems of fractional-order partial differential equations. In this paper, we study shifted Jacobi polynomials in the case of two variables and develop some new operational matrices of fractional-order integrations as well as fractional-order differentiations. By the use of these operational matrices, we present a new and easy method for solving a generalized class of coupled systems of fractionalorder partial differential equations subject to some initial conditions. We convert the system under consideration to a system of easily solvable algebraic equation without discretizing the system, and obtain a highly accurate solution. Also, the proposed method is compared with some other well-known differential transform methods. The proposed method is computer oriented. We use MatLab to perform the necessary calculation. The next two parts will appear soon.

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Existence and uniqueness of solutions for fractional order m-point boundary value problems

Cited by 11 publications

References 10 publications

Application of topological degree method in quantitative behavior of fractional differential equations

Application of topological degree method in quantitative behavior of fractional differential equations

Existence results and Hyers-Ulam stability to a class of nonlinear arbitrary order differential equations

A generalized scheme based on shifted Jacobi polynomials for numerical simulation of coupled systems of multi-term fractional-order partial differential equations

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