Boundary Value Problems Of Fractional Order Research Articles

Positive Solutions for a Fractional Boundary Value Problem with Lidstone Like Boundary Conditions

We consider a higher order fractional boundary value problem with Lidstone like boundary conditions, where the nonlinearity is an L1-Carathèodory function. We first consider the lower order problem. Then, by using a convolution to construct the Green’s function for the higher order problem, we are able to apply a recent fixed point theorem to show the existence of positive solutions of the boundary value problem.

Study of Fractional-Order Boundary Value Problems Using Nondiscretization Numerical Method

This paper is devoted to present a numerical scheme based on operational matrices to compute approximate solutions to fractional-order boundary value problems. For the mentioned operational matrices, we utilize fractional-order Bernoulli polynomials. Since fractional-order problems are usually difficult to treat for their corresponding analytical or exact solutions, therefore, we need sophisticated methods to find their best numerical solutions. The presented numerical scheme has the ability to reduce the proposed problem to the corresponding algebraic equations. The obtained algebraic equations are then solved by using the computational software MATLAB for the corresponding numerical results. The used method has the ability to save much more time and also is reliable to secure the proper amount of memory. Several examples are solved by using the considered method. Also, the solutions are compared with their exact solution graphically. In addition, the absolute errors for different scale values are presented graphically. In addition, we compare our results with the results of the shifted Legendre polynomials spectral method.

Some results on fractional Hahn difference boundary value problems

Abstract Fractional Hahn boundary value problems are significant tools to describe mathematical and physical phenomena depending on non-differentiable functions. In this work, we develop certain aspects of the theory of fractional Hahn boundary value problems involving fractional Hahn derivatives of the Caputo type. First, we construct the Green function for an α th \alpha {\rm{th}} -order fractional boundary value problem, with 1 < α < 2 1\lt \alpha \lt 2 , and discuss some important properties of the Green function. The solutions to the proposed problems are obtained in terms of the Green function. The uniqueness of the solutions is proved by various fixed point theorems. The Banach’s contraction mapping theorem, the Schauder’s theorem, and the Browder’s theorem are used.

Existence of solutions to the iterative system of nonlinear two-point tempered fractional order boundary value problems

Existence of solutions to the iterative system of nonlinear two-point tempered fractional order boundary value problems

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.

On Novel Mathematical Modeling for Studying a Class of Nonlinear Caputo-Type Fractional-Order Boundary Value Problems Emerging in CGT

Chemical graph theory (CGT) is a field of mathematical science that applies classical graph theory to chemical structures and processes. Chemical graphs are the principal data format used in cheminformatics to illustrate chemical interactions. Several researchers have addressed boundary-value problems using star graphs. Star graphs were used since their method requires a central point linked to other vertices but not to itself. Our objective is to expand the mechanism by introducing the idea of an isobutane graph that has the chemical formula C4H10 and CAS number 75-28-5. By using the appropriate fixed point theory findings, this paper investigates the existence of solutions to fractional boundary value problems of Caputo type on such graphs. Additionally, two examples are provided to strengthen our important conclusions.

Existence, Uniqueness, and Stability of Solutions to Variable Fractional Order Boundary Value Problems

This paper investigates the sufficient conditions for the existence and uniqueness of a class of Riemann-Liouville fractional differential equations of variable order with fractional boundary conditions. The problem is converted into differential equations of constant orders by combining the concepts of generalized intervals and piecewise constant functions. We derive the required conditions for ensuring the uniqueness of the problem in order to utilize the Banach fixed point theorem. The stability of the obtained solution in the Ulam-Hyers-Rassias (UHR) sense is also investigated, and we finally provide an illustrative example.

Existence Results for an m-Point Mixed Fractional-Order Problem at Resonance on the Half-Line

This work considers the existence of solutions for a mixed fractional-order boundary value problem at resonance on the half-line. The Mawhin’s coincidence degree theory will be used to prove existence results when the dimension of the kernel of the linear fractional differential operator is equal to two. An example is given to demonstrate the main result obtained.

Some novel analysis of two different Caputo-type fractional-order boundary value problems

Nowadays, a number of classical order results are being analyzed in the sense of fractional derivatives. In this research work, we discuss two different boundary value problems. In the first half of the paper, we generalize an integer-order boundary value problem into fractional-order and then we demonstrate the existence and uniqueness of the solution subject to the Caputo fractional derivative. First, we recall some results and then justify our main results with the proofs of the given theorems. We conclude our results by presenting an illustrative example. In the other half of the paper, we extend the Banach's contraction theorem to prove the existence and uniqueness of the solution to a sequential Caputo fractional-order boundary value problem.

Numerical Solutions of Non-Local Problem for fractional order differential equations

The paper focuses on the study of problem- related to the linear second order boundary value problem (BVP). Three generalization styles are discussed. In the first, the second order BVP with non-local condition of the Dirichlet type is studied. In the second, the fractional order BVP is considered. In the third, the fractional order BVP with non-local condition is treated. The finite difference representation of the BVP is employed to reduce the problem into a system of algebraic equations, in addition, the implicit- explicit treatment to the nonlocal problems is introduced. In the implicit- explicit treatment the computational work is reduced considerably.

Countably infinitely many positive solutions for iterative system of singular R-L fractional order bvp with R-S integral boundary conditions

In this article, we derive sufficient conditions for the existence of countably infinitely many positive solutions for an iterative system of higher order singular Rimean- Liouville(R-L) fractional order boundary value problems with Riemann–Stieltjes(R-S) integral boundary conditions involving increasing homeomorphism and positive homomorphism operator(IHPHO) by applying Hölder’s inequality and Krasnoselskii’s cone fixed point theorem in a Banach space. Also, we derive sufficient conditions for the uniqueness of the solution for the problem by fixed point theorem on a complete metric space.

3D Boundary Element Model for Ultrasonic Wave Propagation Fractional Order Boundary Value Problems of Functionally Graded Anisotropic Fiber-Reinforced Plates

This paper proposes a three–dimensional (3D) local boundary element model based on meshless moving least squares (MLS) method for ultrasonic wave propagation fractional order boundary value problems of functionally graded anisotropic (FGA) fiber-reinforced plates. The problem domain is split into several circular sub-domains. The nodal points are randomly distributed across the examined region. Each node is the focal point of a circular sub-domain that encircles it. The Laplace-transform approach is used to solve dynamic issues. In the local weak form of the governing equations for the converted quantities, a unit test function is utilized. The Gauss divergence theorem to the weak-form is used to produce local boundary-domain integral equations. A meshless approximation is achieved using the MLS method. To find time-dependent solutions, an inverse Laplace-transform approach is used. The effects of the fractional order parameter, functionally graded material, anisotropy, and the time characteristic of the laser pulse are investigated. The proposed method’s validity and performance are demonstrated for a two-dimensional problem with excellent agreement with the finite element method.

Darbo Fixed Point Criterion on Solutions of a Hadamard Nonlinear Variable Order Problem and Ulam-Hyers-Rassias Stability

The existence aspects along with the stability of solutions to a Hadamard variable order fractional boundary value problem are investigated in this research study. Our results are obtained via generalized intervals and piecewise constant functions and the relevant Green function, by converting the existing Hadamard variable order fractional boundary value problem to an equivalent standard Hadamard fractional boundary problem of the fractional constant order. Further, Darbo’s fixed point criterion along with Kuratowski’s measure of noncompactness is used in this direction. As well as, the Ulam-Hyers-Rassias stability of the proposed Hadamard variable order fractional boundary value problem is established. A numerical example is presented to express our results’ validity.

An Application of Homotopy Perturbation Method to Fractional-Order Thin Film Flow of the Johnson–Segalman Fluid Model

Thin film flow is an important theme in fluid mechanics and has many industrial applications. These flows can be observed in oil refinement process, laser cutting, and nuclear reactors. In this theoretical study, we explore thin film flow of non-Newtonian Johnson–Segalman fluid on a vertical belt in fractional space in lifting and drainage scenarios. Modelled fractional-order boundary value problems are solved numerically using the homotopy perturbation method along with Caputo definition of fractional derivative. In this study, instantaneous and average velocities and volumetric flux are computed in lifting and drainage cases. Validity and convergence of homotopy-based solutions are confirmed by finding residual errors in each case. Moreover, the consequences of different fractional and fluid parameters are graphically studied on the velocity profile. Analysis shows that fractional parameters have opposite effects of the fluid velocity.

Ulam–Hyers Stability and Uniqueness for Nonlinear Sequential Fractional Differential Equations Involving Integral Boundary Conditions

Fractional-order boundary value problems are used to model certain phenomena in chemistry, physics, biology, and engineering. However, some of these models do not meet the existence and uniqueness required in the mainstream of mathematical processes. Therefore, in this paper, the existence, stability, and uniqueness for the solution of the coupled system of the Caputo-type sequential fractional differential equation, involving integral boundary conditions, was discussed, and investigated. Leray–Schauder’s alternative was applied to derive the existence of the solution, while Banach’s contraction principle was used to examine the uniqueness of the solution. Moreover, Ulam–Hyers stability of the presented system was investigated. It was found that the theoretical-related aspects (existence, uniqueness, and stability) that were examined for the governing system were satisfactory. Finally, an example was given to illustrate and examine certain related aspects.

An infinite number of nonnegative solutions for iterative system of singular fractional order Boundary value problems

In this paper, we consider the iterative system of singular Rimean-Liouville fractional-order boundary value problems with Riemann-Stieltjes integral boundary conditions involving increasing homeomorphism and positive homomorphism operator(IHPHO). By using Krasnoselskii’s cone fixed point theorem in a Banach space, we derive sufficient conditions for the existence of an infinite number of nonnegative solutions. The sufficient conditions are also derived for the existence of a unique nonnegative solution to the addressed problem by fixed point theorem in complete metric space. As an application, we present an example to illustrate the main results.

Denumerably many positive solutions for fractional order boundary value problems

Denumerably many positive solutions for fractional order boundary value problems

New similarity variable to transform the fluid flow from PDEs into fractional-order ODEs: Numerical study

The recent research aims to study the fractional-order differential equations of the higher-order. The problems consist of momentum and thermal boundary layers. The similarity variables are introduced to convert the partial differential equations (PDEs) into the fractional-order ordinary differential equations (ODEs). The two nonlinear problems of fractional order ODEs are tackled through the (Fractional Differential equation-12) FDE-12 method. The obtained results show that the fractional-order problems are more nonlinear as compared to the integer-order differential equations. The fractional-order increasing the nonlinearity and consequently more efforts required to obtain the outputs. The impact of the physical parameters in the case of the integer-order and fractional-order are obtained and discussed. It is observed that the impact of the physical parameters more compact in the case of fractional order as compared to the integer-order case. The important features of the drag force and heat transfer rate in the case of the fractional-order boundary value problem also have been analyzed and discussed.

An efficient numerical approach for solving two-point fractional order nonlinear boundary value problems with Robin boundary conditions

This article proposes new strategies for solving two-point Fractional order Nonlinear Boundary Value Problems (FNBVPs) with Robin Boundary Conditions (RBCs). In the new numerical schemes, a two-point FNBVP is transformed into a system of Fractional order Initial Value Problems (FIVPs) with unknown Initial Conditions (ICs). To approximate ICs in the system of FIVPs, we develop nonlinear shooting methods based on Newton’s method and Halley’s method using the RBC at the right end point. To deal with FIVPs in a system, we mainly employ High-order Predictor–Corrector Methods (HPCMs) with linear interpolation and quadratic interpolation (Nguyen and Jang in Fract. Calc. Appl. Anal. 20(2):447–476, 2017) into Volterra integral equations which are equivalent to FIVPs. The advantage of the proposed schemes with HPCMs is that even though they are designed for solving two-point FNBVPs, they can handle both linear and nonlinear two-point Fractional order Boundary Value Problems (FBVPs) with RBCs and have uniform convergence rates of HPCMs, mathcal{O}(h^{2}) and mathcal{O}(h^{3}) for shooting techniques with Newton’s method and Halley’s method, respectively. A variety of numerical examples are demonstrated to confirm the effectiveness and performance of the proposed schemes. Also we compare the accuracy and performance of our schemes with another method.

Existence and Uniqueness of Positive Solutions for System of (p,q,r)-Laplacian Fractional Order Boundary Value Problems

In this paper the existence of unique positive solutions for system of (p,q,r)-Lapalacian Sturm-Liouville type two-point fractional order boundary vaue problems is established by an application of n-fixed point theorem of ternary operators on partially ordered metric spaces.