Three-point Boundary Value Conditions Research Articles

Qualitative properties and approximate solutions of thermostat fractional dynamics system via a nonsingular kernel operator

PurposeThis paper aims to study qualitative properties and approximate solutions of a thermostat dynamics system with three-point boundary value conditions involving a nonsingular kernel operator which is called Atangana-Baleanu-Caputo (ABC) derivative for the first time. The results of the existence and uniqueness of the solution for such a system are investigated with minimum hypotheses by employing Banach and Schauder's fixed point theorems. Furthermore, Ulam-Hyers (UH) stability, Ulam-Hyers-Rassias UHR stability and their generalizations are discussed by using some topics concerning the nonlinear functional analysis. An efficiency of Adomian decomposition method (ADM) is established in order to estimate approximate solutions of our problem and convergence theorem is proved. Finally, four examples are exhibited to illustrate the validity of the theoretical and numerical results.Design/methodology/approachThis paper considered theoretical and numerical methodologies.FindingsThis paper contains the following findings: (1) Thermostat fractional dynamics system is studied under ABC operator. (2) Qualitative properties such as existence, uniqueness and Ulam–Hyers–Rassias stability are established by fixed point theorems and nonlinear analysis topics. (3) Approximate solution of the problem is investigated by Adomain decomposition method. (4) Convergence analysis of ADM is proved. (5) Examples are provided to illustrate theoretical and numerical results. (6) Numerical results are compared with exact solution in tables and figures.Originality/valueThe novelty and contributions of this paper is to use a nonsingular kernel operator for the first time in order to study the qualitative properties and approximate solution of a thermostat dynamics system.

\u03b1-\u03c8-contractions and solutions of a q-fractional differential inclusion with three-point boundary value conditions via computational results

We review the existence of solutions for a three-point nonlinear q-fractional differential equation and also its related inclusion. In this way, we use α-ψ-contractions and multifunctions. Also, we provide two examples to illustrate our main results. Finally by providing some algorithms and tables, we give some numerical computations for the results.

On the new fractional hybrid boundary value problems with three-point integral hybrid conditions

We investigate some new class of hybrid type fractional differential equations and inclusions via some nonlocal three-point boundary value conditions. Also, we provide some examples to illustrate our results.

Existence of positive solution to the boundary value problems for coupled system of nonlinear fractional differential equations

In this paper, we investigate the existence criteria of at least one positive solution to the three-point boundary value problems with coupled system of Riemann-Liouville type nonlinear fractional order differential equations. The analysis of this study is based on the well-known Schauder's fixed point theorem. Some new existence and multiplicity results for coupled system of Riemann-Liouville type nonlinear fractional order differential equation with three-point boundary value conditions are obtained.

The Existence and Asymptotic Estimates of Solutions for a Third-Order Nonlinear Singularly Perturbed Boundary Value Problem

In this paper, we consider a class of third-order nonlinear differential equation with singular perturbation subject to three-point boundary value conditions, whose solution exhibits a boundary layer at one endpoint. By using the Schauder fixed point theorem, Green’s function and the method of upper–lower solutions, we first establish an existence result of corresponding boundary value problem without perturbation. Furthermore, by constructing an appropriate lower solution-upper solution pair, as well as analysis technique, the existence and asymptotic estimates of the solutions for the singularly perturbed boundary value problems are obtained.

Existence and approximation of solution for a nonlinear second-order three-point boundary value problem

A nonlinear second-order ordinary differential equation with four cases of three-point boundary value conditions is studied by investigating the existence and approximation of solutions. First, the integration method is proposed to transform the considered boundary value problems into Hammerstein integral equations. Second, the existence of solutions for the obtained Hammerstein integral equations is analyzed by using the Schauder fixed point theorem. The contraction mapping theorem in Banach spaces is further used to address the uniqueness of solutions. Third, the approximate solution of Hammerstein integral equations is constructed by using a new numerical method, and its convergence and error estimate are analyzed. Finally, some numerical examples are addressed to verify the given theorems and methods.

Existence and multiplicity of positive solutions for a class of fractional differential equations with three-point boundary value conditions

In this paper, we consider the nonlinear three-point boundary value problem of fractional differential equations $$D^{\alpha}_{0^{+}}u(t)+a(t)f\bigl(t,u(t)\bigr)=0, \quad 0< t< 1, 2< \alpha\leq3, $$ with boundary conditions $$u(0)=0,\qquad D^{\beta}_{0^{+}}u(0)=0,\qquad D^{\beta}_{0^{+}}u(1)=bD^{\beta}_{0^{+}}u( \xi),\quad 1\leq\beta\leq2, $$ involving Riemann-Liouville fractional derivatives $D^{\alpha}_{0^{+}}$ and $D^{\beta}_{0^{+}}$ , where $a(t)$ maybe singular at $t=0$ or $t=1$ . We use the Banach contraction mapping principle and the Leggett-Williams fixed point theorem to obtain the existence and uniqueness of positive solutions and the existence of multiple positive solutions. We investigate the above fractional differential equations without many preconditions by the fixed point index theory and obtain the existence of a single positive solution. Some examples are given to show the applicability of our main results.

Approximate Solution of Bagley–Torvik Equations with Variable Coefficients and Three-point Boundary-value Conditions

The fractional Bagley–Torvik equation with variable coefficients is investigated under three-point boundary-value conditions. By using the integration method, the considered problems are transformed into Fredholm integral equations of the second kind. It is found that when the fractional order is $$1<\alpha <2,$$ the obtained Fredholm integral equation is with a weakly singular kernel. When the fractional order is $$0<\alpha <1,$$ the given Fredholm integral equation is with a continuous kernel or a weakly singular kernel depending on the applied boundary-value conditions. The uniqueness of solution for the obtained Fredholm integral equation of the second kind with weakly singular kernel is addressed in continuous function spaces. A new numerical method is further proposed to solve Fredholm integral equations of the second kind with weakly singular kernels. The approximate solution is made and its convergence and error estimate are analyzed. Several numerical examples are computed to show the effectiveness of the solution procedures.

On a Time-Fractional Integrodifferential Equation via Three-Point Boundary Value Conditions

The existence and the uniqueness theorems play a crucial role prior to finding the numerical solutions of the fractional differential equations describing the models corresponding to the real world applications. In this paper, we study the existence of solutions for a time-fractional integrodifferential equation via three-point boundary value conditions.

Bifurcation from interval and positive solutions of the three-point boundary value problem for fractional differential equations

This paper investigates the existence of positive solutions for fractional differential equations with the three-point boundary value conditions. The main tools used here are bifurcation techniques and topological degree theory. Two examples to illustrate the applications of main results are also given.

On a class of singular boundary value problems with singular perturbation

In this paper we investigate a class of singular second order differential equations with singular perturbation subject to three-point boundary value conditions, whose solution exhibits a couple of boundary layers at two endpoints. We first establish a lower–upper solutions theorem by using the Schauder fixed point theorem. By the asymptotic expansions and the lower–upper solutions theorem we obtain the existence, asymptotic estimates and uniqueness for the proposed problem. Several examples are given for illustrating our results.

Solvability of p-Laplace equations subject to three-point boundary value problems

In this paper, the existences of solutions for p-Laplace equations subject to three-point boundary value conditions at resonance and non-resonance are studied by using degree theory and some known results are improved.