Nonlinear Multi-point Boundary Value Problems Research Articles

Study of multilayer flow of a bi-viscous Bingham fluid sandwiched between hybrid nanofluid in a vertical slab with nonlinear Boussinesq approximation

Bi-viscosity Bingham plastic fluids are used to understand the rheological characteristics of pigment–oil suspensions, polymeric gels, emulsions, heavy oil, etc. In many industrial and engineering problems involving high-temperature situation, a linear density-temperature variation is inadequate to describe the convective heat transport. Therefore, the characteristics of the nonlinear convective flow of a bi-viscous Bingham fluid (BVBF) through three layers in a vertical slab are studied. The two outer layers of the oil-based hybrid nanofluid and the intermediate layer of BVBF are considered. The thermal buoyancy force is governed by the nonlinear Boussinesq approximation. Continuity of heat flux, velocity, shear stress, and temperature are imposed on the interfaces. The governing equations are derived from the Navier–Stokes equation, conservation of energy, and conservation of mass for three layers. The nonlinear multi-point (four-point) boundary value problem is solved using the differential transform method (DTM). Converging DTM solutions are obtained, and they are validated. The entropy equation and Bejan number were also derived and analyzed. It is established that the nonlinear density–temperature variation leads to a significant improvement in the magnitude of the velocity and temperature profiles due to the increased buoyancy force, and as a result, the drag force on the walls gets reduced. The drag force on the slab gets reduced by decreasing the volume fraction of nanoparticles. Furthermore, nonlinear convection and mixed convection give rise to an advanced rate of heat transport on the walls and thereby to an enhanced heat transport situation.

Higher Order Nonlinear Multi-Point Fractional Boundary Value Problems

Higher Order Nonlinear Multi-Point Fractional Boundary Value Problems

Solving Multi-Point Boundary Value Problems Using Sinc-Derivative Interpolation

In this paper, the Sinc-derivative collocation method is used to solve linear and nonlinear multi-point boundary value problems. This is done by interpolating the first derivative of the unknown variable via Sinc numerical methods and obtaining the desired solution through numerical integration of the interpolation and all higher order derivatives through successive differentiation of the interpolation. Non-homogeneous boundary conditions are reduced to homogeneous using suitable transformations. The efficiency and the accuracy of the method are tested using illustrative examples previously considered by other researchers who used different approaches. The results show the excellent performance of the Sinc-derivative collocation method.

Existence of multiple solutions for nonlinear multi-point boundary value problems

In this paper, we study some nonlinear second order multi-point boundary value problems. We first give a lemma about the characteristic values of the corresponding linear operator. Then, by fixed point theorems in the recent existing literature, we obtain the existence of multiple solutions for these nonlinear second order multi-point boundary value problems, including two positive solutions, two negative solutions, and one sign-changing solution.

Xistence of positive solutions for nonlinear multipoint p-Laplacian dynamicequations on time scales

In this paper, we investigate the existence of positive solutions for nonlinear multipoint boundary value problems for p-Laplacian dynamic equations on time scales with the delta derivative of the nonlinear term. Sufficient assumptions are obtained for existence of at least twin or arbitrary even positive solutions to some boundary value problems. Our results are achieved by appealing to the fixed point theorems of Avery-Henderson. As an application, an example to demonstrate our results is given.

A method of solving a nonlinear boundary value problem with a parameter for a loaded differential equation

A nonlinear loaded differential equation with a parameter on a finite interval is studied. The interval is partitioned by the load points, at which the values of the solution to the equation are set as additional parameters. A nonlinear boundary value problem for the considered equation is reduced to a nonlinear multipoint boundary value problem for the system of nonlinear ordinary differential equations with parameters. For fixed parameters, we obtain the Cauchy problems for ordinary differential equations on the subintervals. Substituting the values of the solutions to these problems into the boundary condition and continuity conditions at the partition points, we compose a system of nonlinear algebraic equations in parameters. A method of solving the boundary value problem with a parameter is proposed. The method is based on finding the solution to the system of nonlinear algebraic equations composed.

An Iterative Approach for the Numerical Solution of Fractional BVPs

In this article, an efficient numerical scheme is presented for the numerical solution of nonlinear multi-point boundary value problems. This method is a combination of Laplace transform and a decomposition procedure, commonly known as the Laplace Decomposition Method. This semi-analytical iterative scheme is highly accurate locally as it produces series solutions, which converges over a relatively short interval. In order to demonstrate the efficiency and simplicity of the method, the proposed strategy is implemented for a number of boundary value problems of fractional order. Comparisons with other existing techniques are reported. Few iterates of this iterative scheme are needed to obtain highly accurate solution, which indicates that the rate of convergence of the scheme is relatively high.

Nonlinear scalar multipoint boundary value problems at resonance

ABSTRACTIn this work we provide conditions for the existence of solutions to nonlinear boundary value problems of the form subject to for . The existence of solutions will be proved under a mild growth condition on the nonlinearity, g, which must hold only on a bounded subset of .

Reproducing kernel pseudospectral method for the numerical investigation of nonlinear multi-point boundary value problems

This paper presents a computational method to solve nonlinear boundary value problems with multi-point boundary conditions. These problems have important applications in the theoretical physics and engineering problems. The method is based on reproducing kernel Hilbert spaces operational matrices and an iterative technique is used to overcome the nonlinearity of the problem. Furthermore, a rigorous convergence analysis is provided and some numerical tests reveal the high efficiency and versatility of the proposed method. The results of numerical experiments are compared with analytical solutions and the best results reported in the literature to confirm the good accuracy of the presented method.

Existence results for a coupled system of nonlinear fractional multi-point boundary value problems at resonance

By using the coincidence degree theory, we present an existence result for a coupled system of nonlinear fractional differential equations with multi-point boundary conditions at resonance.

Positive solutions of nonlinear multi-point boundary value problems

This paper deals with the existence of positive solutions of nonlinear differential equation $$\begin{aligned} u^{\prime \prime }(t)+ a(t) f(u(t) )=0,\quad 0<t <1, \end{aligned}$$ subject to the boundary conditions $$\begin{aligned} u(0)=\sum _{i=1}^{m-2} a_i u (\xi _i) ,\quad u^{\prime } (1) = \sum _{i=1}^{m-2} b_i u^{\prime } (\xi _i), \end{aligned}$$ where $$ \xi _i \in (0,1) $$ with $$ 0< \xi _1<\xi _2< \cdots<\xi _{m-2} < 1,$$ and $$a_i,b_i $$ satisfy $$a_i,b_i\in [0,\infty ),~~ 0< \sum _{i=1}^{m-2} a_i <1,$$ and $$ \sum _{i=1}^{m-2} b_i <1. $$ By using Schauder’s fixed point theorem, we show that it has at least one positive solution if f is nonnegative and continuous. Positive solutions of the above boundary value problem satisfy the Harnack inequality $$\begin{aligned} \displaystyle \inf _{0 \le t \le 1} u(t) \ge \gamma \Vert u\Vert _\infty . \end{aligned}$$

Existence of positive solutions for a third-order multipoint boundary value problem and extension to fractional case

In this paper, we study a nonlinear third-order multipoint boundary value problem by the monotone iterative method. We then obtain the existence of monotone positive solutions and establish iterative schemes for approximating the solutions. In addition, we extend the considered problem to the Riemann-Liouville-type fractional analogue. Finally, we give a numerical example for demonstrating the efficiency of the theoretical results.

SOME EXISTENCE RESULTS OF SEMILINEAR SINGULARLY PERTURBED NONLOCAL BOUNDARY VALUE PROBLEMS

By standard barrier solution method associated with Schauder fixed point theorem, we establish an existence theory for nonlinear second order nonlinear multi-point boundary value problem $(1.3)$, $(1.4)$. Through the pervious existence theorem, we mainly work out some asymptotic behaviors of solutions for semilinear singularly perturbed three-point boundary value problem $(1.1)$, $(1.2)$. Barrier solutions will be constructed explicitly when the boundary or interior layers occur, respectively.

Positive Solutions for Multipoint Boundary Value Problems for Singular Fractional Differential Equations

A class of nonlinear multipoint boundary value problems for singular fractional differential equations is considered. By means of a coupled fixed point theorem on ordered sets, some results on the existence and uniqueness of positive solutions are obtained.

Existence of Positive Solutions for a Nonlinear Higher-Order Multipoint Boundary Value Problem

We study the existence of positive solutions for a nonlinear higher-order multipoint boundary value problem. By applying a monotone iterative method, some existence results of positive solutions are obtained. The main result is illustrated with an example.

Numeric treatment of nonlinear second order multi-point boundary value problems using ANN, GAs and sequential quadratic programming technique

In this paper, computational intelligence technique are presented for solving multi - point nonlinear boundary value problems based on artificial neural networks, evolutionary computing approach, and active-set technique. The neural network is to provide convenient methods for obtaining useful model based on unsupervised error for the differential equations. The motivation for presenting this work comes actually from the aim of introducing a reliable framework that combines the powerful features of ANN optimized with soft computing frameworks to cope with such challenging system. The applicability and reliability of such methods have been monitored thoroughly for various boundary value problems arises in science, engineering and biotechnology as well. Comprehensive numerical experimentations have been performed to validate the accuracy, convergence, and robustness of the designed scheme. Comparative studies have also been made with available standard solution to analyze the correctness of the proposed scheme.

Existence of positive solutions for nonlocal boundary value problem of fractional differential equation

Abstract In this paper, we study a type of nonlinear fractional differential equations multi-point boundary value problem with fractional derivative in the boundary conditions. By using the upper and lower solutions method and fixed point theorems, some results for the existence of positive solutions for the boundary value problem are established. Some examples are also given to illustrate our results.

New Wavelets Collocation Method for Solving Second-Order Multipoint Boundary Value Problems Using Chebyshev Polynomials of Third and Fourth Kinds

This paper is concerned with introducing two wavelets collocation algorithms for solving linear and nonlinear multipoint boundary value problems. The principal idea for obtaining spectral numerical solutions for such equations is employing third- and fourth-kind Chebyshev wavelets along with the spectral collocation method to transform the differential equation with its boundary conditions to a system of linear or nonlinear algebraic equations in the unknown expansion coefficients which can be efficiently solved. Convergence analysis and some specific numerical examples are discussed to demonstrate the validity and applicability of the proposed algorithms. The obtained numerical results are comparing favorably with the analytical known solutions.

The existence of solutions for nonlinear fractional multipoint boundary value problems at resonance

Abstract A class of nonlinear fractional multipoint boundary value problems at resonance is considered in this article. The existence results are obtained by the method of the coincidence degree theory of Mawhin. An example is given to illustrate the results. MSC:34A08.

APPROXIMATE SOLUTION OF NONLINEAR MULTI-POINT BOUNDARY VALUE PROBLEM ON THE HALF-LINE

In this work, we construct a novel weighted reproducing kernel space and give the expression of reproducing kernel function skillfully. Based on the orthogonal basis established in the reproducing kernel space, an efficient algorithm is provided to solve the nonlinear multi-point boundary value problem on the half-line. Uniformly convergence of the approximate solution and convergence estimation of our algorithm are studied. Numerical results show our method has high accuracy and efficiency.