Journal ArticleDOI
Differential equations of fractional order:methods results and problem —I
TLDR
A survey of the methods and results in the theory of such ordinary fractional differential equations is given in this article, where the Laplace transform, operational calculas compositional methods for the solution of linear differential equations of fractional order are discussed.Abstract:
Thc paper deals with the so-called differential equations of fractional order in which an unknown function is contained under the operation of a derivative of fractional order. A survey of the methods and results in the theory of such ordinary fractional differential equations is given. In particular, the method based on the reduction of the Cauchy-type problem for the fractional differential equations to the Volterra integral equations is discussed, and the Laplace transform, operational calculas compositional methods for the solution of linear differential equations of fractional order are presented. Problems and new trends of research are discussed.read more
Citations
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Journal ArticleDOI
Positive solutions for boundary value problem of nonlinear fractional differential equation
Zhanbing Bai,Haishen Lü +1 more
TL;DR: In this paper, the positive solution of nonlinear fractional difier- ential equation with semi-positive nonlinearity was investigated and the existence results of positive solution were obtained by using Krasnosel'skii flxed point theorem.
Book ChapterDOI
Multi-index Mittag-Leffler Functions
TL;DR: In this paper, Dzherbashian [Dzh60] defined a function with positive α 1 > 0, α 2 > 0 and real α 1, β 2, β 3, β 4, β 5, β 6, β 7, β 8, β 9, β 10, β 11, β 12, β 13, β 14, β 15, β 16, β 17, β 18, β 20, β 21, β 22, β 24
Journal ArticleDOI
Algorithms for the fractional calculus: A selection of numerical methods
TL;DR: In this article, a collection of numerical algorithms for the solution of various problems arising in fractional models is presented, which will give the engineer the necessary tools required to work with fractional model in an efficient way.
Journal ArticleDOI
Solving nonlinear fractional partial differential equations using the homotopy analysis method
TL;DR: In this paper, the homotopy analysis method is applied to solve nonlinear fractional partial differential equations (FPDE) with initial conditions, which are introduced by replacing some integer-order time derivatives by fractional derivatives, and the results of applying this procedure to the studied cases show the high accuracy and efficiency of the new technique.
Journal ArticleDOI
On positive solutions of a nonlocal fractional boundary value problem
TL;DR: In this paper, the existence and uniqueness of positive solutions for a nonlocal boundary value problem of fractional differential equation was investigated, and the uniqueness of the positive solution was obtained by the use of contraction map principle and some Lipschitz type conditions.
References
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Book
An Introduction to the Fractional Calculus and Fractional Differential Equations
Kenneth S. Miller,Bertram Ross +1 more
TL;DR: The Riemann-Liouville Fractional Integral Integral Calculus as discussed by the authors is a fractional integral integral calculus with integral integral components, and the Weyl fractional calculus has integral components.
Book
Fractional Integrals and Derivatives: Theory and Applications
TL;DR: Fractional integrals and derivatives on an interval fractional integral integrals on the real axis and half-axis further properties of fractional integral and derivatives, and derivatives of functions of many variables applications to integral equations of the first kind with power and power-logarithmic kernels integral equations with special function kernels applications to differential equations as discussed by the authors.
Proceedings Article
Stability results for fractional differential equations with applications to control processing
TL;DR: In this article, stability results for finite-dimensional linear fractional differential systems in state-space form are given for both internal and external stability, and the main qualitative result is that stabilities are guaranteed iff the roots of some polynomial lie outside the closed angular sector.