Nonlinear Fractional Boundary Value Problem Research Articles

Ulam stability of solutions of a discrete boundary value problem with fractional order

In this present work, we investigate Ulam - Hyers stability for the following nonlinear discrete fractional order boundary value problem of the form for , with boundary conditions , where is a continuous and is the Caputo fractional difference operator with 1 < δ ≤ 2.

A New Iterative Method for the Numerical Solution of High-Order Non-linear Fractional Boundary Value Problems

The boundary value problems (BVPs) have attracted the attention of many scientists from both practical and theoretical points of view, for these problems have remarkable applications in different branches of pure and applied sciences. Due to this important property, this research aims to develop an efficient numerical method for solving a class of nonlinear fractional BVPs. The proposed method is free from perturbation, discretization, linearization, or restrictive assumptions, and provides the exact solution in the form of a uniformly convergent series. Moreover, the exact solution is determined by solving only a sequence of linear BVPs of fractional-order. Hence, from practical viewpoint, the suggested technique is efficient and easy to implement. To achieve an approximate solution with enough accuracy, we provide an iterative algorithm that is also computationally efficient. Finally, four illustrative examples are given verifying the superiority of the new technique compared to the other existing results.

Fractional optimal control problems on a star graph: Optimality system and numerical solution

<p style='text-indent:20px;'>In this paper, we study optimal control problems for nonlinear fractional order boundary value problems on a star graph, where the fractional derivative is described in the Caputo sense. The adjoint state and the optimality system are derived for fractional optimal control problem (FOCP) by using the Lagrange multiplier method. Then, the existence and uniqueness of solution of the adjoint equation is proved by means of the Banach contraction principle. We also present a numerical method to find the approximate solution of the resulting optimality system. In the proposed method, the <inline-formula><tex-math id="M1">\begin{document}$ L2 $\end{document}</tex-math></inline-formula> scheme and the Grünwald-Letnikov formula is used for the approximation of the Caputo fractional derivative and the right Riemann-Liouville fractional derivative, respectively, which converts the optimality system into a system of linear algebraic equations. Two examples are provided to demonstrate the feasibility of the numerical method.

Existence of Positive Solutions of Boundary Value Problem For FractionalOrder Differential Equation

Recently boundary value problems for differential equations of non-integral order have studied in many papers ( see [1,2] ). Zaho etal [ 1 ] studied the following boundary value problem of fractional differential equations. Where denotes the Rimann-Liouville fractional derivative equation of order . By using the lower and upper solution method and fixed point theorem. Liang and Zhang [3] studied the non-linear fractional differential boundary value problem Where is a real number . is the Rimann-Liouville fractional differential operator of order . By means of fixed point theorems , they obtained results on the existence of positive solutions for boundary value problem of fractional differential equations. In this paper , we deal with some existence of positive solution of the following non-linear fractional differential equation. Where is a real number. denotes Rimann-Liouville fractional derivative of order . Our work based on Banach contraction mapping and Krasnoel'skii fixed point theorems to investigate the existence of positive solution. Finally , we suggest studing the existence solutions for the following Integrodifferential equation with boundary value conditions Where H is a nonlinear integral operator given as

Existence and Ulam's type stability results for a class of fractional boundary value problems on a star graph

This paper is concerned with a nonlinear fractional boundary value problem on a star graph. By using a transformation, the suggested problem is converted into an equivalent system of fractional boundary value problem. Schaefer's fixed point theorem and Banach's contraction principle is used to establish its existence and uniqueness results. Further, different kinds of Ulam's type stability results for the proposed problem have been discussed. Finally, two examples are presented to illustrate the application of the obtained results.

Numerical solution of fractional Bratu–type BVPs: a generalized variational iteration approach

PurposeThe purpose of this study is to implement a newly introduced numerical scheme for the numerical solution of a class of nonlinear fractional Bratu-type boundary value problems (BVPs).Design/methodology/approachThis strategy is based on a generalization of the variational iteration method (VIM). This proposed generalized VIM (GVIM) is particularly suitable for tackling BVPs.FindingsThis scheme yields accurate solutions for a class of nonlinear fractional Bratu-type BVPs, for which the errors are uniformly distributed across a given domain. A proof of convergence is included. The numerical results confirm that this approach overcomes the deficiency of the VIM and other methods that exist in the literature in the sense that the solution does not deteriorate as the authors move away from the initial starting point.Originality/valueThe method introduced is based on original research that produces new knowledge. To the best of the authors’ knowledge, this is the first time that this GVIM is applied to fractional BVPs.

Eigenvalue Criteria for Existence of Positive Solutions to Fractional Boundary Value Problem

The existence and multiplicity of positive solutions for the nonlinear fractional differential equation boundary value problem (BVP) DC0+αyx+fx,yx=0, 0&lt;x&lt;1, y0=y′1=y″0=0 is established, where 2&lt;α≤3, CD0+α is the Caputo fractional derivative, and f:0,1×0,∞⟶0,∞ is a continuous function. The conclusion relies on the fixed-point index theory and the Leray-Schauder degree theory. The growth conditions of the nonlinearity with respect to the first eigenvalue of the related linear operator is given to guarantee the existence and multiplicity.

The existence of solutions for nonlinear fractional boundary value problem and its Lyapunov-type inequality involving conformable variable-order derivative

We consider a nonlinear fractional boundary value problem involving conformable variable-order derivative with Dirichlet conditions. We prove the existence of solutions to the considered problem by using the upper and lower solutions’ method with Schauder’s fixed-point theorem. In addition, under some assumptions on the nonlinear term, a new Lyapunov-type inequality is given for the corresponding boundary value problem. The obtained inequality provides a necessary condition for the existence of nontrivial solutions to the considered problem and a method to prove uniqueness for the nonhomogeneous boundary value problem. These new results are illustrated through examples.

Existence and uniqueness of monotone positive solutions for fractional higher-order BVPs

We discuss the existence and uniqueness of monotone positive solutions for a class of higher-order nonlinear fractional differential equations infinite-point boundary value problems (for short, BVPs) on cones. The existence and uniqueness of solutions are obtained via applying the properties of the Green function and a fixed point theorem. Our analysis is based on the operator equation T ω + S ω = ω on an ordered Banach space. Finally, a example is given to illustrate our results.

Existence of solutions for a nonlinear fractional p-Laplacian boundary value problem

The main objective of this paper is to prove the existence of solutions for a fractional p-Laplacian boundary value problem containing both left Riemann–Liouville and right Caputo fractional derivatives. The proofs are based on the upper and lower solutions method and Schauder’s fixed point theorem. The paper is ended by a numerical example.

Monotone iterative method for a p-Laplacian boundary value problem with fractional conformable derivatives

By using monotone iterative method, the extremal solutions and the unique solution are obtained for a nonlinear fractional p-Laplacian boundary value problem involving fractional conformable derivatives and nonlocal integral boundary conditions. Comparison theorems related to the proposed study are also proved. The paper concludes with an illustrative example for the main result.

Existence of Solutions of Nonlinear and Non-local Fractional Boundary Value Problems

In this paper, we establish new results for non-local boundary value problems. In particular, we study a fractional differential equation where the associated integral equation has a kernel that is not bounded above and changes its sign, so that, the positive sign of the possible solutions is generally not ensured. We provide some examples which support the theory and illustrate the applicability of the obtained results.

Iterative unique positive solutions for a new class of nonlinear singular higher order fractional differential equations with mixed-type boundary value conditions

In this paper, we consider a new class of singular nonlinear higher order fractional boundary value problems supplemented with sum of Riemann–Stieltjes integral type and nonlocal infinite-point discrete type boundary conditions. The fractional derivative of different orders is involved in the nonlinear terms and boundary conditions, and the nonlinear terms are allowed to be singular in regard to not only time variable but also space variables. A unique positive solution is established by using the fixed point theorem of mixed monotone operator. In addition, some significant properties of the unique solution depending on the parameter λ are stated. In the end, two examples are worked out to illustrate our main results.

An effective numerical method for solving nonlinear variable-order fractional functional boundary value problems through optimization technique

An optimization method standing on a new basis formed by the transcendental Bernstein series (TBS) is proposed. The TBS includes the unknown free coefficients and control parameters for solving nonlinear variable-order fractional functional boundary value problems (NV-FFBVP). The corresponding operational matrices for variable-order fractional derivatives are derived for expanding the solution by means of TBS. The TBS is a generalization of the Bernstein polynomials (BP) and represent a superset of BP. In the particular cases where all the control parameters are zero, the TBS method is equivalent to the BP method. The proposed technique reduces the NV-FFBVP to a system of algebraic equations and, subsequently, to the problem of finding the free coefficients and control parameters using an optimization technique. Several numerical results reveal the computational performance and reliability of the method.

Direct and integrated radial functions based quasilinearization schemes for nonlinear fractional differential equations

In this article, two radial basis functions based collocation schemes, differentiated and integrated methods (DRBF and IRBF), are extended to solve a class of nonlinear fractional initial and boundary value problems. Before discretization, the nonlinear problem is linearized using generalized quasilinearization. An interesting proof via generalized monotone quasilinearization for the existence and uniqueness for fractional order initial value problem is given. This convergence analysis also proves quadratic convergence of the generalized quasilinearization method. Both the schemes are compared in terms of accuracy and convergence and it is found that IRBF scheme handles inherent RBF ill-condition better than corresponding DRBF method. Variety of numerical examples are provided and compared with other available results to confirm the efficiency of the schemes.

Nontrivial Solutions for a Higher Order Nonlinear Fractional Boundary Value Problem Involving Riemann-Liouville Fractional Derivatives

In this paper using topological degree we study the existence of nontrivial solutions for a higher order nonlinear fractional boundary value problem involving Riemann-Liouville fractional derivatives. Here, the nonlinearity can be sign-changing and can also depend on the derivatives of unknown functions.

Existence and uniqueness results for a nonlinear Caputo fractional boundary value problem on a star graph

In this paper, we study a nonlinear Caputo fractional boundary value problem on a star graph. By means of transformation, an equivalent system of fractional boundary value problem is obtained. Then we establish the existence and uniqueness results by fixed point theory: in particular, using Banach's contraction principle and Schaefer's fixed point theorem. We also present some examples to illustrate the application of our results.

Properties of Green\u2019s function and the existence of different types of solutions for nonlinear fractional BVP with a parameter in integral boundary conditions

This paper is concerned with the impact of the parameter on the existence of different types of solutions for a class of nonlinear fractional integral boundary value problems with a parameter that causes the sign of Green’s function associated with the BVP to change. By using the Guo–Krasnoselskii fixed point theorem, the Leray–Schauder nonlinear alternative, and the analytic technique, we give the range of the parameter for the existence of strong positive solutions, strong negative solutions, negative solutions, and sign-changing solutions for the boundary value problem. Some examples are given to illustrate our main results.

Weak Solutions to a System of Nonlinear Fractional Boundary Value Problems Via Variational Form

In this paper, the weak solutions to a class of nonlinear system of fractional boundary value problems are studied. Each equation of the system does have two kinds of nonlinear terms; one of them is in terms of gradient, and the other one is a nonlinear source term with respect to dependent variable. It is proved that there exists at least three distinct weak solutions by using the critical point theory and the variational method. To this aim, we apply the well-known theorem on the construction of the critical set of functionals with a weak compactness condition. Moreover, the main result is demonstrated by some examples to show its validity.

Lyapunov inequalities of left focal q-difference boundary value problems and applications

In this paper, we establish some Lyapunov-type inequalities for a class of linear and nonlinear fractional q-difference boundary value problems under Cauchy boundary conditions. As applications, we use the inequality to obtain an interval, where Mittag-Leffler function has no real zeros. In addition, we also derive nonexistence results for fractional q-difference boundary value problem.