Order Boundary Value Problems Research Articles

On the existence and uniqueness of a nonlinear q-difference boundary value problem of fractional order

In this research collection, we estimate the existence of the unique solution for the boundary value problem of nonlinear fractional [Formula: see text]-difference equation having the given form [Formula: see text] [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] represents the Caputo-type nonclassical [Formula: see text]-derivative of order [Formula: see text]. We use well-known principal of Banach contraction, and Leray–Schauder nonlinear alternative to vindicate the unique solution existence of the given problem. Regarding the applications, some examples are solved to justify our outcomes.

A time–efficient variable shape parameter Kansa–radial basis function method for the solution of nonlinear boundary value problems

In this study we consider the application of a Kansa–radial basis function (RBF) collocation method for solving two– and three–dimensional nonlinear boundary value problems (BVPs) of second and fourth order. In this variable shape parameter approach, a distinct shape parameter is linked with each RBF in the approximation of the solution and the total set of unknowns in the resulting discretized nonlinear problem comprises the RBF coefficients in the approximation and the set of (distinct) shape parameters. The solution of the system of nonlinear equations is achieved using the MATLAB©optimization toolbox functions fsolve or lsqnonlin. Unlike previous applications of these routines to nonlinear BVPs, we exploit the option offered in these functions to provide the analytical expression of the Jacobian of the nonlinear systems in question and show, in several numerical applications, how this leads to spectacular savings in computational time.

Stability analysis of an overdetermined fourth order boundary value problem via an integral identity

We consider an overdetermined fourth order boundary value problem in which the boundary value of the Laplacian of the solution is prescribed, in addition to the homogeneous Dirichlet boundary condition. It is known that, in the case where the prescribed boundary value is a constant, this overdetermined problem has a solution if and only if the domain under consideration is a ball. In this paper, we study the shape of a domain admitting a solution to the overdetermined problem when the prescribed boundary value is slightly perturbed from a constant. We derive an integral identity for the fourth order Dirichlet problem and a nonlinear weighted trace inequality, and the combination of them results in a quantitative stability estimate which measures the deviation of a domain from a ball in terms of the perturbation of the boundary value.

Numerical solutions of higher order boundary value problems via wavelet approach

This paper presents a numerical scheme based on Haar wavelet for the solutions of higher order linear and nonlinear boundary value problems. In nonlinear cases, quasilinearization has been applied to deal with nonlinearity. Then, through collocation approach computing solutions of boundary value problems reduces to solve a system of linear equations which are computationally easy. The performance of the proposed technique is portrayed on some linear and nonlinear test problems including tenth, twelfth, and thirteen orders. Further convergence of the proposed method is investigated via asymptotic expansion. Moreover, computed results have been matched with the existing results, which shows that our results are comparably better. It is observed from convergence theoretically and verified computationally that by increasing the resolution level the accuracy also increases.

An Inverse Problem Solution for Thermal Conductivity Reconstruction

This work deals with the inverse problem of reconstructing the thermal conductivity coefficient of the (2+1)D heat equation from over–posed data at the boundaries. The proposed solution uses a variational approach for identifying the coefficient. The inverse problem is reformulated as a higher–order elliptic boundary–value problem for minimization of a quadratic functional of the original equation. The resulting system consists of a well–posed fourth–order boundary–value problem for the temperature and an explicit equation for the unknown thermal conductivity coefficient. The existence and uniqueness of the resulting higher–order boundary–value problem are investigated. The unique solvability of the inverse coefficient problem is proven. The numerical algorithm is validated and applied to problems of reconstructing continuous nonlinear coefficient and discontinuous coefficients. Accurate and stable numerical solutions are obtained.

Kansa RBF collocation method with auxiliary boundary centres for high order BVPs

In this study we apply a Kansa-radial basis function (RBF) collocation method to 2D and 3D boundary value problems (BVPs) governed by high order partial differential equations (PDEs) of order 2N where N∈N,N≥3. As in such problems there are N boundary conditions (BCs), N distinct sets of boundary centres are needed. These could all be placed on the boundary with each set being different to the other or, alternatively, each set of boundary centres could be placed on a corresponding distinct curve surrounding the boundary of the problem. We apply these approaches to several 2D and 3D high order BVPs.

ASYMPTOTIC DISTRIBUTION OF EIGENVALUES AND EIGENFUNCTIONS OF A NONLOCAL BOUNDARY VALUE PROBLEM

In this work, we obtain asymptotic formulas for eigenvalues and eigenfunctions of the second order boundary-value problem with a Bitsadze–Samarskii type nonlocal boundary condition.

Lidstone\u2013Euler interpolation and related high even order boundary value problem

We consider the Lidstone–Euler interpolation problem and the associated Lidstone–Euler boundary value problem, in both theoretical and computational aspects. After a theorem of existence and uniqueness of the solution to the Lidstone–Euler boundary value problem, we present a numerical method for solving it. This method uses the extrapolated Bernstein polynomials and produces an approximating convergent polynomial sequence. Particularly, we consider the fourth-order case, arising in various physical models. Finally, we present some numerical examples and we compare the proposed method with a modified decomposition method for a tenth-order problem. The numerical results confirm the theoretical and computational ones.

Solution of nonlinear boundary value problem by S-iteration

In this paper we import a novel approach to solve the non-linear fourth order boundary value problem. The basic strategy depends on integral operator equation which includes Green’s function and fixed point iteration. To ensure the method, three illustrations were conferred. From the residual or absolute error calculations, it is shown that the S-iteration for contraction operator gives fast convergence than Krasnoselskii–Mann’s iteration.

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.

Multiple Solutions for a Sixth Order Boundary Value Problem

This work presents conditions for the existence of multiple solutions for a sixth order equation with homogeneous boundary conditions using Avery Peterson's theorem. In addition, non-trivial examples are presented and a new numerical method based on the Banach's Contraction Principle is introduced.

Importance of considering unsaturated triaxial tests including ceramic disk as initial and boundary value problems

Considering unsaturated triaxial tests including a ceramic disk as initial and boundary value problems, a series of suction change, isotropic consolidation, and drained shearing processes was simulated using a soil-water-air coupled elastoplastic finite deformation analysis code. As a result, it was demonstrated that the delayed behavior of a wetting-induced collapse during the suction change and isotropic consolidation processes, as well as water-absorption behavior during the subsequent drained shearing process, were attributed not to the characteristics of the soil specimen, but to the permeability of the disk. Thus, it is important to take the ceramic disk into account in unsaturated triaxial tests regarded as initial and boundary value problems in order to understand the temporal change behavior of unsaturated soil specimens. Otherwise, there is a risk that the behavior appearing as the solution of an initial and boundary value problem may be modeled as a constitutive relation.

Solutions of the variational equation for an n-th order boundary value problem with an integral boundary condition

We discuss differentiation of solutions to the boundary value problem y ( n ) = f ( x , y , y ′ , y ′ ′ , … , y ( n − 1 ) ) , a < x < b , y ( i ) ( x j ) = y i j , 0 ≤ i ≤ m j , 1 ≤ j ≤ k − 1 , y ( i ) ( x k ) + ∫ c d p y ( x ) d x = y i k , 0 ≤ i ≤ m k , ∑ i = 1 k m i = n , with respect to the boundary data. We show that under certain conditions, partial derivatives of the solution y(x) of the boundary value problem with respect to the various boundary data exist and solve the associated variational equation along y(x).

The numerical solution of two point boundary value problem for the Helmholtz type equation by finite difference method with non regular step length between nodes

In this article, we have presented a variable step finite difference method for solving second order boundary value problems in ordinary differential equations. We have discussed the convergence and established that proposed has at least cubic order of accuracy. The proposed method tested on several model problems for the numerical solution. The numerical results obtained for these model problems with known / constructed exact solution confirm the theoretical conclusions of the proposed method. The computational results obtained for these model problems suggest that method is efficient and accurate.

Efficient Numerical Algorithm for the Solution of Eight Order Boundary Value Problems by Haar Wavelet Method.

In this paper, the Haar technique is applied to both nonlinear and linear eight-order boundary value problems. The eight-order derivative in the boundary value problem is approximated using Haar functions in this technique and the integration process is used to obtain the expression of the lower order derivative and the approximate solution of the unknown function. For the verification of validation and convergence of the proposed technique, three linear and two nonlinear examples are taken from the literature. The results are also compared with other methods available in the literature. Maximum absolute and root mean square errors at various collocation and Gauss points are contrasted with the exact solution. The convergence rate is also measured, which is almost equivalent to 2, using different numbers of collocation points.

A new fixed point iteration method for nonlinear third-order BVPs

In this article, we shall present a novel approach based on embedding Green's function into Ishikawa fixed point iterative procedure for the numerical solution of a broad class of boundary value problems of third order. A linear integral operator expressed in terms of Green's function is constructed, then the well-known Ishikawa fixed point iterative scheme is applied to obtain a new iterative scheme. The aim of our alternative strategy is to overcome the major deficiency of other iterative schemes that usually result in the deterioration of the error as the domain increases. Furthermore the proposed strategy will improve the rate of convergence of other existing methods that are based on Picard's and Mann's iterative schemes. Convergence results of the iterative algorithm have been proved. A number of numerical examples shall be solved to illustrate the method and demonstrate its reliability and accuracy. Moreover, we shall compare our results with both the analytical and the numerical solutions obtained by other methods that exist in the literature.

Singular matrices arising in the MFS from certain boundary and pseudo-boundary symmetries

We consider the application of the method of fundamental solutions (MFS) to certain two–dimensional second order boundary value problems (BVPs). We assume that the domain of the problem under consideration is a regular polygon and that the sources in the MFS are placed uniformly on a circle concentric to and surrounding the polygon. For certain uniform distributions of the collocation points we prove that, for general classes of regular polygons, the MFS discretization leads to singular coefficient matrices.

Numerical Solution of Fifth Order Boundary Value Problems by Collocation Method with Sixth Order B-Splines

Collocation method with sixth degree B-splines as basis functions has been developed to solve a fifth order special case boundary value problem. To get an accurate solution by the collocation method with sixth degree B-splines, the original sixth degree B-splines are redefined into a new set of basis functions which in number match with the number of collocation points. The method is tested for solving both linear and nonlinear boundary value problems. The proposed method is giving better results when compared with the methods available in literature.

Applications of new hybrid algorithm based on advanced cuckoo search and adaptive Gaussian quantum behaved particle swarm optimization in solving ordinary differential equations

This article solves first and second order differential equations with initial and/or boundary conditions by transforming these equations into unconstrained/bound constrained optimization problems. In order to solve these problems, a hybrid algorithm based on advanced cuckoo search (CS) algorithm and adaptive Gaussian quantum behaved particle swarm optimization (AGQPSO) is proposed. The CS algorithm is modified first by changing the step size in the simplified version. After that half of the total population is upgraded by this modified CS algorithm and another half is upgraded by AGQPSO algorithm. Then deletion strategy of CS algorithm is applied on the whole updated population. Next, to test the performance of the proposed hybrid algorithm, a number of benchmarks bound constrained optimization problems with different dimensions are considered and solved. Then this algorithm is applied fruitfully in first and second order initial value problems and boundary value problems by expressing the said problems in the form of bound constrained optimization problems.