Fractional Boundary Research Articles

A high-efficient accurate coupled mesh-free scheme for 2D/3D space-fractional convection-diffusion/Burgers' problems

This paper first develops a high-efficient accurate mesh-free scheme (CSPH-SFCD) for multi-dimensional time-dependent spatial fractional convection-diffusion (SFCD) equations based on the Riemann-Liouville (RL) derivative in convex domain by coupling a corrected smoothed particle hydrodynamics (CSPH) method with a numerical integral of RL. Subsequently, for the first application, the proposed CSPH-SFCD coupled with up-wind (UW) scheme (CSPH-SFCD-UW) is tentatively extended to investigate the nonlinear shock-wave phenomena in 2D space-fractional Burgers' problem (SFBP) with fractional Neumann boundary at high Reynolds number. The proposed mesh-free scheme for time-dependent SFCD/SFBP with fractional boundary is mainly motived by: (a) the SFCD model with RL fractional calculus is discretized by employing a numerical integral formula and a CSPH scheme, and then a conditionally stable CSPH-SFCD method is first derived; (b) the UW scheme is introduced to stabilize the numerical investigation of convection-dominated SFBP; (c) the fractional Neumann boundary (FNB) condition can be easily enforced in the proposed discretized scheme, and the integer-order NB condition is treated by employing virtual particles interpolation technique; (d) the MPI parallel computing technique is adopted to enhance the computational efficiency. Firstly, the numerical convergence and stability of the proposed scheme are discussed and illustrated by solving several multi-dimensional benchmarks. Secondly, the cases of two-sided and left-sided RL fractional derivatives are also discussed and demonstrated in the numerical tests. The advantages of the proposed method are demonstrated by simulating the 2D/3D SFCD equations in convex domains or under non-uniform point distribution. Finally, the 2D two-component SFBP with FNBC are numerically investigated by using the proposed method, and compared with the FDM results. The complex nonlinear non-local properties governed by SFBP are numerically predicted by the proposed scheme. All the numerical results show that the proposed mesh-free method for time-dependent spatial SFCD problems is high-efficient, flexible and accurate. • A high-efficient accurate CSPH-SFCD scheme is first developed to 2D/3D SFCD equations. • The CSPH-SFCD-UW scheme is first proposed to investigate shock-wave behavior in 2D SFBP. • The FNB condition can be easily enforced in the proposed approximated scheme. • The advantages of the present scheme are shown by numerical tests on convex domain.

A Lyapunov-type inequality for a fractional differential equation under multi-point boundary conditions

In this paper we discus on a Lyapunov-type inequality for a fractional differential equation involving sequential generalized Caputo fractional derivatives with boundary conditions. The results presented in this paper is new to the corresponding results in the literature.

Free vibration analysis of rotating FGP sandwich cylindrical shells with metal-foam core layer

This study offers a unique solution for the free vibration of rotating FGP sandwich cylindrical shells under different boundary conditions. A new type of sandwich structures with a metal-foam core and two coating FGM layers is investigated. Equations of motion governing the free vibration of rotating shells are derived by the first-order, shear-deformation shell theory. Single-term modal functions are introduced in Rayleigh-Ritz method to efficiently determine natural frequencies of the shell. After result validations, the influence of volume fraction index, distribution schemes, core-layer thickness, porosity coefficient, rotating velocity, geometrical parameters, and boundary conditions on the shell natural frequencies are studied.

Influence of microstructural defects on free flexural vibration of cracked functionally graded plates in thermal medium using XFEM

This paper presents the influence of microstructural defects on the vibration characteristics of the cracked functionally graded plate in a thermal environment. Extended finite element formulation is implemented to model the cracked functionally graded plate based on higher-order shear deformation theory (HSDT). The level set function is implemented to track the crack in the functionally graded plate domain. The crack in the functionally graded plate domain is addressed by enriching the primary variable with additional functions using the partition of unity technique. The power-law distribution is implemented for the gradation of the material properties along the thickness direction to form functionally graded material (FGM). The microstructural defects are assumed as porosities in the plate. The porosity index determines the porosity in the FGM plate. The convergence and comparative study have been performed to verify the current formulation’s efficacy, accuracy, and reliability. Further, various numerical examples are presented. The effects of different influential parameters like porosity, volume fraction index, crack length, thickness ratio, and boundary condition on the natural frequencies have been discussed in detail. The presented computational approach is accurate, efficient, and robust enough to investigate the vibration response of the cracked FGM porous plates, which can provide a reference for future research by researchers.

USER-ORIENTED INTELLIGENCE MINING UNDER THE EXISTENCE OF SOLUTIONS TO INTEGRAL BOUNDARY VALUE PROBLEMS FOR FUZZY PARTIAL FRACTIONAL DIFFERENTIAL EQUATIONS

In order to accurately perceive user intention and improve the reliability of user intention information mining, the existence of solution of integral boundary value problem of fuzzy partial fractional differential equation is used to mine the information of user intention. Firstly, the periodic boundary value problem of fractional order fuzzy linear differential equation, periodic boundary value problem of fractional order fuzzy nonlinear differential equation and periodic boundary value problem of fractional order fuzzy coupled differential system are described in this research. The existence of the solution of the integral boundary value problem of the fuzzy partial fractional order differential equation is proved. Then a user-oriented information mining framework and an evaluation model based on the existence of solutions for the integral boundary value problem of fuzzy partial fractional differential equations are constructed. Finally, this research makes a case study and a comprehensive evaluation of user-oriented information mining based on the existence of solutions of integral boundary value problems for fuzzy partial fractional differential equations. The results show that the method based on the existence of the solution of the integral boundary value problem of fuzzy partial fractional order differential equation is feasible and scientific for the information mining of user intention. It is also concluded that this method is suitable for the modelling and solving of intelligence mining with complicated and unclear user intention. The research provides a good guidance for information mining oriented to user intention.

EXISTENCE AND UNIQUENESS SOLUTION FOR THREE-POINT HADAMARD-TYPE FRACTIONAL VOLTERRA BVP

In this paper we study the existence and uniqueness solution for a first kind fractional Volterra boundary value problem involving Hadamard type and three-point boundary conditions. Our analysis is based on Krasnoselskii-Zabreiko’s fixed point theorem and Banach contraction principle. As an application we discuss a Hadamard type boundary value problem with fractional integral boundary conditions. We emphasize that our results are new in the context of Hadamard fractional calculus and are well illustrated with the aid of examples.

Existence results of sequential fractional Caputo sum-difference boundary value problem

<abstract><p>In this article, we study the existence and uniqueness results for a sequential nonlinear Caputo fractional sum-difference equation with fractional difference boundary conditions by using the Banach contraction principle and Schaefer's fixed point theorem. Furthermore, we also show the existence of a positive solution. Our problem contains different orders and four fractional difference operators. Finally, we present an example to display the importance of these results.</p></abstract>

On a coupled nonlinear fractional integro-differential equations with coupled non-local generalised fractional integral boundary conditions

On a coupled nonlinear fractional integro-differential equations with coupled non-local generalised fractional integral boundary conditions

On a coupled nonlinear fractional integro-differential equations with coupled non-local generalised fractional integral boundary conditions

On a coupled nonlinear fractional integro-differential equations with coupled non-local generalised fractional integral boundary conditions

Fractional N-Laplacian boundary value problems with jumping nonlinearities in the fractional Orlicz\u2013Sobolev spaces

We investigate the multiplicity of solutions for problems involving the fractional N-Laplacian. We obtain three theorems depending on the source terms in which the nonlinearities cross some eigenvalues. We obtain these results by direct computations with the eigenvalues and the corresponding eigenfunctions for the fractional N-Laplacian eigenvalue problem in the fractional Orlicz–Sobolev spaces, the contraction mapping principle on the fractional Orlicz–Sobolev spaces and Leray–Schauder degree theory.

Compact and Noncompact Solutions to Generalized Sturm–Liouville and Langevin Equation with Caputo–Hadamard Fractional Derivative

In this work, through using the Caputo–Hadamard fractional derivative operator with three nonlocal Hadamard fractional integral boundary conditions, a new type of the fractional-order Sturm–Liouville and Langevin problem is introduced. The existence of solutions for this nonlinear boundary value problem is theoretically investigated based on the Krasnoselskii in the compact case and Darbo fixed point theorems in the noncompact case with aiding the Kuratowski’s measure of noncompactness. To demonstrate the applicability and validity of the main gained findings, some numerical examples are included.

On the generalized fractional snap boundary problems via G-Caputo operators: existence and stability analysis

This research is conducted for studying some qualitative specifications of solution to a generalized fractional structure of the standard snap boundary problem. We first rewrite the mathematical model of the extended fractional snap problem by means of the mathbb{G}-operators. After finding its equivalent solution as a form of the integral equation, we establish the existence criterion of this reformulated model with respect to some known fixed point techniques. Then we analyze its stability and further investigate the inclusion version of the problem with the help of some special contractions. We present numerical simulations for solutions of several examples regarding the fractional mathbb{G}-snap system in different structures including the Caputo, Caputo–Hadamard, and Katugampola operators of different orders.

Existence and uniqueness solutions for nonlinear fractional differential equations with fractional integral boundary conditions

The theory of Schuder fixed point and Banach contraction theorems are used to explain the behavior of solution for differential equation of fractional order β ∈ (2, 3] with fractional integral limit conditions given by .We prove the existence and uniqueness solution of the above equations.

Some Results on Fractional m-Point Boundary Value Problems

In this paper, we will apply some fixed-point theorems to discuss the existence of solutions for fractional m-point boundary value problems D 0 + q u ″ t = h t f u t , t ∈ 0 , 1 , 1 < q ≤ 2 , u ′ 0 = u ″ 0 = u 1 = 0 , u ″ 1 − ∑ i = 1 m − 2 α i u ‴ ξ i = 0 . In addition, we also present Lyapunov’s inequality and Ulam-Hyers stability results for the given m-point boundary value problems.

On a nonlinear sequential four-point fractional q-difference equation involving q-integral operators in boundary conditions along with stability criteria

In this paper, we consider a nonlinear sequential q-difference equation based on the Caputo fractional quantum derivatives with nonlocal boundary value conditions containing Riemann–Liouville fractional quantum integrals in four points. In this direction, we derive some criteria and conditions of the existence and uniqueness of solutions to a given Caputo fractional q-difference boundary value problem. Some pure techniques based on condensing operators and Sadovskii’s measure and the eigenvalue of an operator are employed to prove the main results. Also, the Ulam–Hyers stability and generalized Ulam–Hyers stability are investigated. We examine our results by providing two illustrative examples.

Transient Vibroacoustic Analysis of Functionally Graded Plates

Abstract This paper presents the vibroacoustic response of pure functionally graded (FG) plates under transient loading of mechanical nature. The functionally graded plate is modeled using the conventional first-order shear deformation theory (FSDT) to incorporate the effects of transverse shear and rotary inertia. The mid-surface variables are determined using the finite element method. Transient structural response is determined using Newmark Beta time marching scheme and the acoustic pressure in the free field is obtained using the time-domain Rayleigh integral. The effective material properties of the FG plate and the transient response of both the structural and acoustic fields have been computed in matlab. The influence of the volume fraction index, thickness ratio, and boundary conditions of pure FG plate on its transient vibroacoustic response is investigated by a detailed parametric study.

Existence and Hyers-Ulam stability results of nonlinear implicit Caputo-Katugampola fractional differential equations with p–Laplacian operator

The existence, uniqueness and the Hyers-Ulam stability of solutions for nonlinear p–Laplacian implicit Caputo-Katugampola fractional differential equations (FDEs) with multi-point generalized fractional integral boundary conditions are established. With the aid of the Leray-Schauder nonlinear alternative and the Banach contraction principle, the desired results are obtained. An example is set forth to make efficient the main results.

Non-Trivial Solutions of Non-Autonomous Nabla Fractional Difference Boundary Value Problems

In this article, we present a two-point boundary value problem with separated boundary conditions for a finite nabla fractional difference equation. First, we construct an associated Green’s function as a series of functions with the help of spectral theory, and obtain some of its properties. Under suitable conditions on the nonlinear part of the nabla fractional difference equation, we deduce two existence results of the considered nonlinear problem by means of two Leray–Schauder fixed point theorems. We provide a couple of examples to illustrate the applicability of the established results.

Hilfer fractional differential inclusions with Erd\xe9lyi\u2013Kober fractional integral boundary condition

In this article, we debate the existence of solutions for a nonlinear Hilfer fractional differential inclusion with nonlocal Erdélyi–Kober fractional integral boundary conditions (FIBC). Both cases of convex- and nonconvex-valued right-hand side are considered. Our obtained results are new in the framework of Hilfer fractional derivative and Erdélyi–Kober fractional integral with FIBC via the fixed point theorems (FPTs) for a set-valued analysis. Some pertinent examples demonstrating the effectiveness of the theoretical results are presented.

Efficient difference method for time-space fractional diffusion equation with Robin fractional derivative boundary condition

In this paper, a numerical method is proposed to solve the time-space fractional diffusion equation with Robin fractional derivative boundary condition. Under the weak regularity assumptions of solution, we present a numerical scheme based on the L1 method for time discretization on graded mesh and the Grunwald-Letnikov formula for spatial discretization on uniform mesh. And a fast scheme for the considered problem is constructed based on the exponential summation approximation of the kernel function t−α. Meanwhile, a detailed analysis of stability and convergence is given. Then, the extrapolation method is applied to the space direction to make it reach the second-order accuracy. Finally, numerical experiments show that the proposed method is effective.