scholarly journals Using Shehu integral transform to solve fractional order Caputo type initial value problems

2019 ◽  
Vol 18 (2) ◽  
pp. 75-83 ◽  
Author(s):  
Sania Qureshi ◽  
Prem Kumar
Keyword(s):  
Fractional Order ◽  
Initial Value Problems ◽  
Integral Transform ◽  
Initial Value

scholarly journals Intrinsic Discontinuities in Solutions of Evolution Equations Involving Fractional Caputo–Fabrizio and Atangana–Baleanu Operators

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2023
Author(s):  
Christopher Nicholas Angstmann ◽  
Byron Alexander Jacobs ◽  
Bruce Ian Henry ◽  
Zhuang Xu
Keyword(s):  
Initial Value Problems ◽  
Fractional Derivatives ◽  
Evolution Equations ◽  
Integral Transform ◽  
Initial Value ◽  
Intrinsic Nature ◽  
Recent Interest ◽  
Special Cases ◽  
Singular Kernels

There has been considerable recent interest in certain integral transform operators with non-singular kernels and their ability to be considered as fractional derivatives. Two such operators are the Caputo–Fabrizio operator and the Atangana–Baleanu operator. Here we present solutions to simple initial value problems involving these two operators and show that, apart from some special cases, the solutions have an intrinsic discontinuity at the origin. The intrinsic nature of the discontinuity in the solution raises concerns about using such operators in modelling. Solutions to initial value problems involving the traditional Caputo operator, which has a singularity inits kernel, do not have these intrinsic discontinuities.

scholarly journals Fractional order differential inclusions on an unbounded domain with infinite delay

Mathematica ◽  
2020 ◽  
Vol 62 (85) (2) ◽  
pp. 167-178
Author(s):  
Mohamed Helal
Keyword(s):  
Fractional Order ◽  
Initial Value Problems ◽  
Differential Inclusions ◽  
Existence Of Solutions ◽  
Infinite Delay ◽  
Sufficient Conditions ◽  
Fréchet Spaces ◽  
Initial Value ◽  
Nonlinear Alternative ◽  
Hyperbolic Differential Inclusions

We provide sufficient conditions for the existence of solutions to initial value problems, for partial hyperbolic differential inclusions of fractional order involving Caputo fractional derivative with infinite delay by applying the nonlinear alternative of Frigon type for multivalued admissible contraction in Frechet spaces.

Convergence rate estimates for the kernelized predictor corrector method for fractional order initial value problems

2021 ◽  
Vol 24 (6) ◽  
pp. 1879-1898
Author(s):  
Joel A. Rosenfeld ◽  
Warren E. Dixon
Keyword(s):  
Convergence Rate ◽  
Fractional Order ◽  
Basis Function ◽  
Initial Value Problems ◽  
Computation Time ◽  
Linear Interpolation ◽  
Kernel Functions ◽  
Initial Value ◽  
Convergence Rate Estimate ◽  
Predictor Corrector

Abstract This manuscript presents a kernelized predictor corrector (KPC) method for fractional order initial value problems, which replaces linear interpolation with interpolation by a radial basis function (RBF) in a predictor-corrector scheme. Specifically, the class of Wendland RBFs is employed as the basis function for interpolation, and a convergence rate estimate is proved based on the smoothness of the particular kernel selected. Use of the Wendland RBFs over Mittag-Leffler kernel functions employed in a previous iteration of the kernelized method removes the problems encountered near the origin in [11]. This manuscript performs several numerical experiments, each with an exact known solution, and compares the results to another frequently used fractional Adams-Bashforth-Moulton method. Ultimately, it is demonstrated that the KPC method is more accurate but requires more computation time than the algorithm in [4].

scholarly journals Iterative Bernstein splines technique applied to fractional order differential equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Zoltan Satmari
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Fractional Order ◽  
Initial Value Problems ◽  
Spline Approximation ◽  
Initial Value ◽  
Numerical Examples ◽  
Fractional Order Differential Equations ◽  
Banach’S Fixed Point Theorem

<p style='text-indent:20px;'>In this work we will discuss about an approximation method for initial value problems associated to fractional order differential equations. For this method we will use Bernstein spline approximation in combination with the Banach's Fixed Point Theorem. In order to illustrate our results, some numerical examples will be presented at the end of this article.</p>

scholarly journals Special Functions as Solutions to the Euler–Poisson–Darboux Equation with a Fractional Power of the Bessel Operator

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1484
Author(s):  
Azamat Dzarakhohov ◽  
Yuri Luchko ◽  
Elina Shishkina
Keyword(s):  
Initial Value Problems ◽  
Fractional Derivatives ◽  
Integral Transform ◽  
Special Functions ◽  
Fractional Power ◽  
Fundamental Solutions ◽  
Initial Value ◽  
Explicit Formulas ◽  
Bessel Differential Operator ◽  
Darboux Equation

In this paper, we consider fractional ordinary differential equations and the fractional Euler–Poisson–Darboux equation with fractional derivatives in the form of a power of the Bessel differential operator. Using the technique of the Meijer integral transform and its modification, fundamental solutions to these equations are derived in terms of the Fox–Wright function, the Fox H-function, and their particular cases. We also provide some explicit formulas for the solutions to the corresponding initial-value problems in terms of the generalized convolutions introduced in this paper.

scholarly journals Novel Techniques for a Verified Simulation of Fractional-Order Differential Equations

2021 ◽  
Vol 5 (1) ◽  
pp. 17
Author(s):  
Andreas Rauh ◽  
Luc Jaulin
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Initial Value Problems ◽  
Initial Conditions ◽  
Order System ◽  
Interval Methods ◽  
Initial Value ◽  
System Models ◽  
Simulation Techniques ◽  
Software Implementations

Verified simulation techniques have been investigated intensively by researchers who are dealing with ordinary and partial differential equations. Tasks that have been considered in this context are the solution to initial value problems and boundary value problems, parameter identification, as well as the solution of optimal control problems in cases in which bounded uncertainty in parameters and initial conditions are present. In contrast to system models with integer-order derivatives, fractional-order models have not yet gained the same attention if verified solution techniques are desired. In general, verified simulation techniques rely on interval methods, zonotopes, or Taylor model arithmetic and allow for computing guaranteed outer enclosures of the sets of solutions. As such, not only the influence of uncertain but bounded parameters can be accounted for in a guaranteed way. In addition, also round-off and (temporal) truncation errors that inevitably occur in numerical software implementations can be considered in a rigorous manner. This paper presents novel iterative and series-based solution approaches for the case of initial value problems to fractional-order system models, which will form the basic building block for implementing state estimation schemes in continuous-discrete settings, where the system dynamics is assumed as being continuous but measurements are only available at specific discrete sampling instants.

scholarly journals Generalization of Fuzzy Laplace Transforms of Fuzzy Fractional Derivatives about the General Fractional Ordern-1<β

2016 ◽  
Vol 2016 ◽  
pp. 1-13
Author(s):  
Amal Khalaf Haydar ◽  
Ruaa Hameed Hassan
Keyword(s):  
Fractional Order ◽  
Initial Value Problems ◽  
Fractional Derivatives ◽  
Laplace Transforms ◽  
Initial Value ◽  
Caputo Fractional Derivatives

The main aim in this paper is to use all the possible arrangements of objects such thatr1of them are equal to 1 andr2(the others) of them are equal to 2, in order to generalize the definitions of Riemann-Liouville and Caputo fractional derivatives (about order0<β<n) for a fuzzy-valued function. Also, we find fuzzy Laplace transforms for Riemann-Liouville and Caputo fractional derivatives about the general fractional ordern-1<β<nunder H-differentiability. Some fuzzy fractional initial value problems (FFIVPs) are solved using the above two generalizations.

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