Fractional Boundary Value Problems Research Articles

Solvability of Three Point Fractional Boundary Value Problem Using Mittag-Leffler Function

Solvability of Three Point Fractional Boundary Value Problem Using Mittag-Leffler Function

Error estimates of Hermite‐Hadamard type inequalities with respect to a monotonically increasing function

Fractional calculus is used to examine and enhance the concept of calculus in diverse fields of science. In this paper, we establish Hermite‐Hadamard inequalities for composite ‐convex function. The generalized identities are established for Riemann‐type fractional integrals. The explored identities are used to examine error estimates of Hermite‐Hadamard inequalities for ‐convex function concerning a strictly monotone function. Some results that exist in literature are obtained as special cases to our general results. The conclusions of this article may be useful in determining the uniqueness of partial differential equations and fractional boundary value problems.

On a Mixed Nonlinear Fractional Boundary Value Problem with a New Class of Closed Integral Boundary Conditions

On a Mixed Nonlinear Fractional Boundary Value Problem with a New Class of Closed Integral Boundary Conditions

LYAPUNOV TYPE INEQUALITY FOR DISCRETE FRACTIONAL BOUNDARY VALUE PROBLEM

In this paper we consider the discrete fractional boundary value problem. The Green’s function and it’s properties are used to find maximum value of function. With the help of maximum value of the function Lyapunov type inequality is obtain for this problem.

RLC electric circuit model of fractional order: a Green's function approach

This article uses a hybrid approach that combines Green's function and a fixed point scheme to obtain a semi-analytical solution to the RLC electrical circuit's non-integer order differential equation. The fractional differential equation's order is determined by the two parameters α ∈ ( 1 , 2 ] and β ∈ ( 0 , 1 ] . The proposed method is unique because it can be extended to general nonlinear fractional systems. Another highly desirable feature of the proposed approach is that the construction of Green's function depends on both endpoints of the time domain, and hence the produced approximate solutions are highly accurate for fractional boundary value problems. We provide examples with figures and tables to demonstrate the effectiveness of our method in solving electrical circuit problems. TThe obtained approximate solutions are compared with exact solutions or numerical simulations to validate the applicability and accuracy of the proposed scheme.

An extended variational iteration method for fractional BVPs encountered in engineering applications

PurposeThe purpose of this paper is to find approximate solutions for a general class of fractional order boundary value problems that arise in engineering applications.Design/methodology/approachA newly developed semi-analytical scheme will be applied to find approximate solutions for fractional order boundary value problems. The technique is regarded as an extension of the well-established variation iteration method, which was originally proposed for initial value problems, to cover a class of boundary value problems.FindingsIt has been demonstrated that the method yields approximations that are extremely accurate and have uniform distributions of error throughout their domain. The numerical examples confirm the method’s validity and relatively fast convergence.Originality/valueThe generalized variational iteration method that is presented in this study is a novel strategy that can handle fractional boundary value problem more effectively than the classical variational iteration method, which was designed for initial value problems.

Solvability for a Hadamard-type fractional integral boundary value problem

In this paper, we study an integral boundary value problem involving a Hadamard-type fractional differential equation. Using fixed point theory and upper-lower solutions, we present some sufficient conditions to obtain existence theorems of positive solutions for the problem. Examples are provided to illustrate our results.

System of fractional boundary value problems at resonance

In this paper, we investigate the existence of solutions for a system of fractional differential equations at resonance set on the interval [0, 1]. We associate a Dirichlet condition at $$t=0$$ and an integral boundary condition at $$t=1$$ . Our main existence theorem relies on the coincidence degree theory. A detailed study of the linear operators involved in the fixed point formulation is carried out. Finally, an example is included to illustrate the applicability of theoretical result.

A method for solving Caputo–Hadamard fractional initial and boundary value problems

In this article, we proposed a method by generalizing the classical CAS wavelets for the approximate solutions of nonlinear fractional Caputo–Hadamard initial and boundary value problems. We have generated the new operational matrices for the generalized CAS (gCAS) wavelets, and these matrices are successfully utilized for the solution of nonlinear Caputo–Hadamard fractional initial and boundary value problems. The method that we have proposed in the present study is the combination of gCAS wavelets (based on new operational matrices) and the quasilinearization technique. We have derived and constructed the gCAS wavelets, the gCAS wavelets operational matrix of Hadamard fractional integral of order , and the gCAS wavelets operational matrix of Hadamard fractional integral for boundary value problems. We have also derived the orthonormality condition for the gCAS wavelets. The purpose of these operational matrices is to make the calculations faster. Furthermore, we worked out the error analysis of the proposed method. We presented the procedure of implementation for both nonlinear Caputo–Hadamard fractional initial and boundary value problems. Numerical simulations are provided to illustrate the reliability and accuracy of the method. Since the Hadamard differential equation is a new and emerging field, many engineers can utilize the proposed method for the numerical simulations of their linear/nonlinear Caputo–Hadamard fractional differential models.

A Comparison Result for the Nabla Fractional Difference Operator

This article establishes a comparison principle for the nabla fractional difference operator ∇ρ(a)ν, 1<ν<2. For this purpose, we consider a two-point nabla fractional boundary value problem with separated boundary conditions and derive the corresponding Green’s function. I prove that this Green’s function satisfies a positivity property. Then, I deduce a relatively general comparison result for the considered boundary value problem.

Stability of solutions to fractional differential equations with time-delays

This paper deals with a fractional boundary value problem involving variable delays. Sufficient conditions for the existence of a unique solution are investigated. Moreover the stability of the unique solution is discussed. A numerical example that emphasizes the importance of the results obtained in this article is also included.

On The Solutions of Fractional Boundary Value Problems for a Combined Caputo Fractional Differential Equations

In this paper, a new class of fractional boundary value problem with the combined Caputo derivative is proposed and the physical interpretation of this new derivative has been explained. Under some assumptions, the positive solutions of the fractional differential equation with the help of Leray-Schauder and Krasnoselskii's fixed point theorems in a cone have been investigated. Moreover, the solution of the fractional Maxwell models involving the combined Caputo derivative by using the extended Laplace transform is obtained. Finally, some examples are given to support theoretical findings.

Approximate solution for a fractional BVP under Ψ‐Riemann–Liouville operators via iterative method and artificial neural networks

In this work, we discussed the existence and uniqueness of the solution for a fractional boundary value problem involving the ‐Riemann–Liouville operators where we used Banach's fixed point theorem; after that, we applied an iterative method and artificial neural networks technique to solve our main problem. Finally, the approximate solution was plotted.

Lyapunov Inequalities for Two Dimensional Fractional Boundary-Value Problems with Mixed Fractional Derivatives

We consider two types of partial fractional differential equations in two dimensions with mixed fractional derivatives. Appropriate Lyapunov-type inequalities are proved, and applications to the certain eigenvalue problems are presented. Moreover, some connections with the fractional variational problems are highlighted.

Existence of positive solutions for singular p-Laplacian Hadamard fractional differential equations with the derivative term contained in the nonlinear term

In this paper, based on the properties of Green function and the eigenvalue of a corresponding linear operator, the existence of positive solutions is investigated by spectral analysis for a infinite-points singular p-Laplacian Hadamard fractional differential equation boundary value problem, and an example is given to demonstrate the validity of our main results.

Fractional boundary value problem in complex domain

Fractional calculus of complex order in complex domain has emerged as a brand-new area of study. Over the past few years, fractional boundary value problems (FBVP) in real variables have been extensively studied but there are few attempts on these types of problems in complex variables. In this study, the existence and uniqueness for the solutions of fractional differential equation (FDE) in complex domain with boundary conditions is examined. We established the existence of solutions using the Krasnoselskii fixed point theorem; however, the uniqueness result is proved by applying the Banach contraction principle. To explain our findings, an illustrative example is presented. The special cases of the derived findings are equivalent to the theorems that already exist.

Fractional Sobolev spaces and Boundary Value Problems via Hadamard derivative

Fractional Sobolev spaces and Boundary Value Problems via Hadamard derivative

$\alpha$-$(h,e)$-convex operators and applications for Riemann-Liouville fractional differential equations

In this paper, we consider a class of $\alpha$-$(h,e)$-convex operators defined in set $P_{h,e}$ and applications with $\alpha> 1$. Without assuming the operator to be completely continuous or compact, by employing cone theory and monotone iterative technique, we not only obtain the existence and uniqueness of fixed point of $\alpha$-$(h,e)$-convex operators, but also construct two monotone iterative sequences to approximate the unique fixed point. At last, we investigate the existence-uniqueness of a nontrivial solution for Riemann-Liouville fractional differential equations integral boundary value problems by employing $\alpha$-$(h,e)$-convex operators fixed point theorem.

Rayleigh–Faber–Krahn, Lyapunov and Hartmann–Wintner Inequalities for Fractional Elliptic Problems

In this paper, in the cylindrical domain, we consider a fractional elliptic operator with Dirichlet conditions. We prove, that the first eigenvalue of the fractional elliptic operator is minimised in a circular cylinder among all cylindrical domains of the same Lebesgue measure. This inequality is called the Rayleigh–Faber–Krahn inequality. Also, we give Lyapunov and Hartmann–Wintner inequalities for the fractional elliptic boundary value problem.

Stability of p(·)-Integrable Solutions for Fractional Boundary Value Problem via Piecewise Constant Functions

The goal of this work is to study a multi-term boundary value problem (BVP) for fractional differential equations in the variable exponent Lebesgue space (Lp(·)). Both the existence, uniqueness, and the stability in the sense of Ulam–Hyers are established. Our results are obtained using two fixed-point theorems, then illustrating the results with a comprehensive example.