Nonlocal Initial Value Problems For Nonlinear Neutral Pantograph Equations With Hilfer-Hadamard Fractional Derivative

2021 ◽  
Vol 10 (1) ◽  
pp. 111-119
Keyword(s):  
Fractional Derivative ◽  
Initial Value Problems ◽  
Initial Value ◽  
Hadamard Fractional Derivative

L 1-solutions of the initial value problems for implicit differential equations with Hadamard fractional derivative

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Benoumran Telli ◽  
Mohammed Said Souid
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Point Theorem ◽  
Initial Value Problems ◽  
Initial Value ◽  
Hadamard Fractional Derivative ◽  
Implicit Differential Equations ◽  
Banach Contraction

Abstract In this paper, we study the existence of integrable solutions for initial value problems for fractional order implicit differential equations with Hadamard fractional derivative. Our results are based on Schauder’s fixed point theorem and the Banach contraction principle fixed point theorem.

Maximum principles and applications for fractional differential equations with operators involving Mittag-Leffler function

2021 ◽  
Vol 24 (4) ◽  
pp. 1220-1230
Author(s):  
Mohammed Al-Refai
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Initial Value Problems ◽  
Maximum Principles ◽  
Initial Value ◽  
Derivative Operator ◽  
Fractional Differential ◽  
Linear And Nonlinear ◽  
Two Parameters

Abstract In this paper, we formulate and prove two maximum principles to nonlinear fractional differential equations. We consider a fractional derivative operator with Mittag-Leffler function of two parameters in the kernel. These maximum principles are used to establish a pre-norm estimate of solutions, and to derive certain uniqueness and positivity results to related linear and nonlinear fractional initial value problems.

scholarly journals Analysis of Multiterm Initial Value Problems with Caputo–Fabrizio Derivative

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Mohammed Al-Refai ◽  
Muhammed Syam
Keyword(s):  
Laplace Transform ◽  
Fractional Derivative ◽  
Initial Value Problems ◽  
Existence Of Solutions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Initial Value ◽  
The Laplace Transform ◽  
Necessary And Sufficient

In this paper, we discuss the solvability of a class of multiterm initial value problems involving the Caputo–Fabrizio fractional derivative via the Laplace transform. We derive necessary and sufficient conditions to guarantee the existence of solutions to the problem. We also obtain the solutions in closed forms. We present two examples to illustrate the validity of the obtained results.

scholarly journals Applications of Optimal Spline Approximations for the Solution of Nonlinear Time-Fractional Initial Value Problems

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 249
Author(s):  
Enza Pellegrino ◽  
Francesca Pitolli
Keyword(s):  
Numerical Methods ◽  
Fractional Derivative ◽  
Initial Value Problems ◽  
Real Life ◽  
Collocation Methods ◽  
Approximation Properties ◽  
Initial Value ◽  
Numerical Tests ◽  
Fractional Differential ◽  
Life Phenomena

Nonlinear fractional differential equations are widely used to model real-life phenomena. For this reason, there is a need for efficient numerical methods to solve such problems. In this respect, collocation methods are particularly attractive for their ability to deal with the nonlocal behavior of the fractional derivative. Among the variety of collocation methods, methods based on spline approximations are preferable since the approximations can be represented by local bases, thereby reducing the computational load. In this paper, we use a collocation method based on spline quasi-interpolant operators to solve nonlinear time-fractional initial value problems. The numerical tests we performed show that the method has good approximation properties.

scholarly journals Initial Value Problems of Linear Equations with the Dzhrbashyan–Nersesyan Derivative in Banach Spaces

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1058
Author(s):  
Vladimir E. Fedorov ◽  
Marina V. Plekhanova ◽  
Elizaveta M. Izhberdeeva
Keyword(s):  
Fractional Derivative ◽  
Banach Spaces ◽  
Initial Value Problems ◽  
Bounded Operator ◽  
Linear Equations ◽  
Initial Boundary ◽  
Inhomogeneous Equation ◽  
Initial Value ◽  
Caputo Derivatives ◽  
Initial Boundary Value

Among the many different definitions of the fractional derivative, the Riemann–Liouville and Gerasimov–Caputo derivatives are most commonly used. In this paper, we consider the equations with the Dzhrbashyan–Nersesyan fractional derivative, which generalizes the Riemann–Liouville and the Gerasimov–Caputo derivatives; it is transformed into such derivatives for two sets of parameters that are, in a certain sense, symmetric. The issues of the unique solvability of initial value problems for some classes of linear inhomogeneous equations of general form with the fractional Dzhrbashyan–Nersesyan derivative in Banach spaces are investigated. An inhomogeneous equation containing a bounded operator at the fractional derivative is considered, and the solution is presented using the Mittag–Leffler functions. The result obtained made it possible to study the initial value problems for a linear inhomogeneous equation with a degenerate operator at the fractional Dzhrbashyan–Nersesyan derivative in the case of relative p-boundedness of the operator pair from the equation. Abstract results were used to study a class of initial boundary value problems for equations with the time-fractional Dzhrbashyan–Nersesyan derivative and with polynomials in a self-adjoint elliptic differential operator with respect to spatial variables.

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