Mujeeb ur Rehman

Existence and uniqueness of solutions for multi-point boundary value problems for fractional differential equations

In this work, we investigate existence and uniqueness of solutions for a class of nonlinear multi-point boundary value problems for fractional differential equations. Our analysis relies on the Schauder fixed point theorem and the Banach contraction principle.

A note on boundary value problems for a coupled system of fractional differential equations

In this work, we establish sufficient conditions for the existence of solutions to a general class of multi-point boundary value problems for a coupled system of fractional differential equations. The differential operator is taken in the Riemann-Liouville sense. By means of a fixed point theorem, existence results for the solutions are established. We include an example to show the applicability of our results.

Three point boundary value problems for nonlinear fractional differential equations

Abstract In this paper, we study existence and uniqueness of solutions to nonlinear three point boundary value problems for fractional differential equation of the type D δ 0 + c u ( t ) = f ( t , u ( t ) , D σ 0 + c u ( t ) ) , t ∈ [ 0 , T ] , u ( 0 ) = α u ( η ) , u ( T ) = β u ( η ) , where 1 l δ l 2 , 0 l σ l 1 , α l 1 , α , β ∈ ℝ , η ∈ ( 0 , T ) , α η ( 1 − β ) + ( 1 − α ) ( T − β η ) ≠ 0 and D c δ 0 + , D c σ 0 + are the Caputo fractional derivatives. We use Schauder fixed point theorem and contraction mapping principle to obtain existence and uniqueness results. Examples are also included to show the applicability of our results.

Existence and uniqueness of solutions for impulsive fractional differential equations

Motivated by some recent developments in the existence theory of impulsive fractional differential equations, in this paper we present a general method for converting an impulsive fractional differential equation to an equivalent integral equation. The applicability of the method is demonstrated by considering some boundary value problems for impulsive fractional differential equations.

Haar wavelet-quasilinearization technique for fractional nonlinear differential equations

In this article, numerical solutions of nonlinear ordinary differential equations of fractional order by the Haar wavelet and quasilinearization are discussed. Quasilinearization technique is used to linearize the nonlinear fractional ordinary differential equation and then the Haar wavelet method is applied to linearized fractional ordinary differential equations. In each iteration of quasilinearization technique, solution is updated by the Haar wavelet method. The results are compared with the results obtained by the other technique and with exact solution. Several initial and boundary value problems are solved to show the applicability of the Haar wavelet method with quasilinearization technique.

Hermite Wavelet Method for Fractional Delay Differential Equations

We proposed a method by utilizing method of steps and Hermite wavelet method, for solving the fractional delay differential equations. This technique first converts the fractional delay differential equation to a fractional nondelay differential equation and then applies the Hermite wavelet method on the obtained fractional nondelay differential equation to find the solution. Several numerical examples are solved to show the applicability of the proposed method.

Haar wavelet Picard method for fractional nonlinear partial differential equations

In this article, we present a solution method for fractional nonlinear partial differential equation. The proposed technique utilizes the Haar wavelets operational matrices method in conjunction with Picard technique. The operational matrices are derived and utilized for the solution of fractional nonlinear partial differential equations. Convergence analysis for the proposed technique has also been given. Numerical examples are provided to illustrate the efficiency and accuracy of the technique.

Modified Chebyshev wavelet methods for fractional delay-type equations

In this article, we develop the Chebyshev wavelet method for solving the fractional delay differential equations and integro-differential equations. According to the development, we approximate the delay unknown functions by the Chebyshev wavelets series at delay time, which we call the delay Chebyshev wavelet series. We also proposed a technique by combining the method of steps and Chebyshev wavelet method for solving fractional delay differential equations. This technique converts the fractional delay differential equation on a given interval to a fractional non-delay differential equation over that interval, by using the function defined on previous interval, and then apply the Chebyshev wavelet method on the obtained fractional non-delay differential equation to find the solution. Numerical examples will be presented to demonstrate the benefits of computing with these approaches.

Wavelet-Galerkin Quasilinearization Method for Nonlinear Boundary Value Problems

A numerical method is proposed by wavelet-Galerkin and quasilinearization approach for nonlinear boundary value problems. Quasilinearization technique is applied to linearize the nonlinear differential equation and then wavelet-Galerkin method is implemented to linearized differential equations. In each iteration of quasilinearization technique, solution is updated by wavelet-Galerkin method. In order to demonstrate the applicability of proposed method, we consider the various nonlinear boundary value problems.

A note on terminal value problems for fractional differential equations on infinite interval

Abstract In this note we establish sufficient conditions for existence and uniqueness of solutions of terminal value problems for a class of fractional differential equations on infinite interval. Some illustrative examples are comprehended to demonstrate the proficiency and utility of our results.