Solution Of Boundary Value Problem Research Articles

A Three-Step Iterative Scheme Based on Green's Function for the Solution of Boundary Value Problems

In this manuscript, we suggest a three-step iterative scheme for finding approximate numerical solutions to boundary value problems (BVPs) in a Banach space setting. The underlying strategy of the scheme is based on embedding Green’s function into the three-step M-iterative scheme, which we will call in the paper M-Green’s iterative scheme. We assume certain possible mild conditions to prove the convergence and stability results of the suggested scheme. We also prove numerically that our M-Green iterative scheme is more effective than the corresponding Mann-Green and Khan-Green iterative schemes. Our results improve and extend some recent results in the literature of Green’s function based iteration schemes.

A novel (2+1)-dimensional nonlinear evolution equation for weakly stratified free-surface boundary layers

To get an insight into the dynamics of the oceanic surface boundary layer we develop an asymptotic model of the nonlinear dynamics of linearly decaying three-dimensional long-wave perturbations in weakly stratified boundary-layer flows. Although in nature the free-surface boundary layers in the ocean are often weakly stratified due to solar radiation and air entrainment caused by wave breaking, weak stratification has been invariably ignored. Here, we consider an idealized hydrodynamic model, where finite-amplitude three-dimensional perturbations propagate in a horizontally uniform unidirectional weakly stratified shear flow confined to a boundary layer adjacent to the water surface. Perturbations satisfy the no-stress boundary condition at the surface. They are assumed to be long compared with the boundary-layer thickness. Such perturbations have not been studied even in a linear setting. By exploiting the assumed smallness of nonlinearity, wavenumber, viscosity and the Richardson number, on applying triple-deck asymptotic scheme and multiple-scale expansion, we derive in the distinguished limit a novel essentially two-dimensional nonlinear evolution equation, which is the main result of the work. The equation represents a generalization of the two-dimensional Benjamin–Ono equation modified by the explicit account of viscous effects and new dispersion due to weak stratification. It describes perturbation dependence on horizontal coordinates and time, while its vertical structure, to leading order, is given by an explicit analytical solution of the linear boundary value problem. It shows the principal importance of weak stratification for three-dimensional perturbations.

Novel Approximate Solutions for Nonlinear Initial and Boundary Value Problems

This paper investigates an effective computational method (ECM) based on the standard polynomials used to solve some nonlinear initial and boundary value problems appeared in engineering and applied sciences. Moreover, the effective computational methods in this paper were improved by suitable orthogonal base functions, especially the Chebyshev, Bernoulli, and Laguerre polynomials, to obtain novel approximate solutions for some nonlinear problems. These base functions enable the nonlinear problem to be effectively converted into a nonlinear algebraic system of equations, which are then solved using Mathematica®12. The improved effective computational methods (I-ECMs) have been implemented to solve three applications involving nonlinear initial and boundary value problems: the Darcy-Brinkman-Forchheimer equation, the Blasius equation, and the Falkner-Skan equation, and a comparison between the proposed methods has been presented. Furthermore, the maximum error remainder () has been computed to prove the proposed methods' accuracy. The results convincingly prove that ECM and I-ECMs are effective and accurate in obtaining novel approximate solutions to the problems.

Inverse scattering problem for a third-order operator with local potential

Inverse scattering problem for the operator representing sum of the operator of the third derivative on semi-axis and of the operator of multiplication by a real function is studied in this paper. Properties of Jost solutions of such an operator are studied and it is shown that these Jost solutions are solutions of the Riemann boundary value problem on a system of rays. The main system of linear singular integral equations is derived. This system is equivalent to the solution of inverse scattering problem.

Numerical solution of third order singular boundary value problems with nonclassical SE-sinc-collocation and nonclassical DE-sinc-collocation

In this paper, we use the nonclassical SE-sinc-collocation and nonclassical DE-sinc-collocation methods for the numerical solution of singular third-order boundary value problems. The novelty of the approach is based on using the new non-classical weight function for sinc method instead of the classic ones. The convergence and error estimation of our methods is discussed. Several examples are solved and numerical results are compared with the existed methods, the obtained results demonstrate the validity of the obtained theoretical results and the efficiency of the our methods.

Variational iteration algorithm for numerical solutions of sixth and seventh order boundary value problems using shifted Vieta-Lucas polynomials

This study employs Shifted Vieta-Lucas Polynomials using the variational iteration approach to numerically resolve sixth and seventh order Boundary Value Problems (BVPs), The proposed method in the study is used, with the trial functions for the approximation being the shifted Vieta Lucas polynomials generated for the given boundary value problem. The study provides an efficient and accurate solutions to higher-order BVPs using a combination of Shifted Vieta-Lucas Polynomials and the variational iteration approach and finds applications in mathematics and other fields of sciences and engineering.

Determining the Spectrum of Eigenvalues and Eigenfunctions for the Bernoulli–Euler Equation with Variable Coefficients by the Peano Method

The paper considers the problem of determining the natural frequencies and eigenwaves of transverse vibrations for the Bernoulli–Euler equation with variable coefficients. Such problems arise both in the case of complex geometry of a vibrating solid and in the case of functionally graded materials or the accumulation of damage in a classical elastic material. Solutions of boundary value problems are constructed using the expansion in Peano series. Under broad assumptions, the uniform convergence of Peano series is shown and estimates of the residual terms are obtained. Examples of the numerical implementation of the proposed procedure are given for bending vibrations of a rod with certain parameters of a variable cross section (geometric heterogeneity) and elastic modulus distribution (physical heterogeneity). Numerical examples are focused on assessing the geometric and elastic properties of samples in an experimental study of the fatigue strength of alloys during high-frequency cyclic tests based on the general principle of point resonant loading. The method proposed for solving problems of resonant vibrations for the Bernoulli–Euler equation can be used in the design of new promising cyclic test schemes and mathematical modeling of fatigue failure processes under high-frequency resonant vibrations.

Boundary corrections on multi-dimensional PDEs

Two new boundary correction techniques are proposed in order to mitigate the order reduction phenomenon associated with the numerical solution of initial boundary value problems for parabolic partial differential equations in arbitrary spatial dimensions with time-dependent Dirichlet boundary conditions. The new techniques are based on the idea of discretizing the PDE problem at the boundary points as similar as possible to that of the interior points of the domain. These new techniques are considered for the time integration with W-methods based on approximate matrix factorization. By suitably modifying the internal stages of the methods on the boundary points, it is illustrated by numerical testing with time-dependent boundary conditions that the new boundary correction techniques are able to keep the same accuracy and order of convergence that the method reaches in the case of homogeneous boundary conditions.

Global structure of positive solutions for a fourth-order boundary value problem with singular data

Global structure of positive solutions for a fourth-order boundary value problem with singular data

NONLINEAR INVERSE BOUNDARY VALUE PROBLEM FOR A SECOND ORDER PARABOLIC EQUATION WITH NONLOCAL BOUNDARY CONDITIONS

The paper studies the classical solvability of an inverse boundary value problem for a second order parabolic equation with nonlocal boundary conditions. For this purpose, first, the considered problem is reduced to an auxiliary equivalent problem in a certain sense. Then, using the Fourier method the auxiliary problem is presented as a system of integral equations. Further, by means of the contraction mappings principle the unique existence of the solution of the obtained system of integral equations is shown. At the end of investigation the existence and uniqueness theorem for the classical solution of the original inverse boundary value problem is proved based on the equivalence of these problems

Existence and Uniqueness of Variable-Order φ-Caputo Fractional Two-Point Nonlinear Boundary Value Problem in Banach Algebra

Using variable-order fractional derivatives in differential equations is essential. It enables more precise modeling of complex phenomena with varying memory and long-range dependencies, improving our ability to describe real-world processes reliably. This study investigates the properties of solutions for a two-point boundary value problem associated with φ-Caputo fractional derivatives of variable order. The primary objectives are to establish the existence and uniqueness of solutions, as well as explore their stability through the Ulam-Hyers concept. To achieve these goals, Banach’s and Krasnoselskii’s fixed point theorems are employed as powerful mathematical tools. Additionally, we provide numerical examples to illustrate results and enhance comprehension of theoretical findings. This comprehensive analysis significantly advances our understanding of variable-order fractional differential equations, providing a strong foundation for future research. Future directions include exploring more complex boundary value problems, studying the effects of varying fractional differentiation orders, extending the analysis to systems of equations, and applying these findings to real-world scenarios, all of which promise to deepen our understanding of Caputo fractional differential equations with variable order, driving progress in both theoretical and applied mathematics.

Использование Python для численного решения начально-краевых задач уравнения теплопроводности и визуализации результатов

в работе описана реализация численного метода контрольного объема для решения одного класса трехмерных начально-краевых задач теплопроводности в шаре на языке программирования Python. Проведено тестирование на задаче с  аналитическим решением и выполнена визуализация результатов расчета средствами Python. this study uses the control volume method for solving a class of 3D initial-boundary value problems of the heat equation in a ball with Python. The code was tested with an analytically solvable problem, and the results were visualized.

The Bulk Trace FEM for the simultaneous solution of structural membranes on all level sets over a bulk domain

AbstractA finite element method for the simultaneous solution of boundary value problems (BVPs) posed on all level sets over some bulk domain is proposed. A level‐set function is used to define ‐dimensional curved manifolds which are bounded by the d‐dimensional bulk domain. The BVPs considered herein model structural ropes and membranes in a large displacement context, resulting in a geometrically non‐linear mechanical model. For the numerical analysis, the bulk domain is discretized using d‐dimensional finite elements which do not conform to the level sets implying the domains of interest. However, they conform to the boundary of the bulk domain. The proposed variant of the FEM is a hybrid between classical and non‐conforming FEMs and may be labeled Bulk Trace FEM. Numerical studies confirm optimal higher‐order convergence rates for the proposed method when simultaneously approximating BVPs on all level sets.

Existence of sign-changing solutions for a third-order boundary value problem with nonlocal conditions of integral type

We prove the existence of at least one sign-changing solution for a third-order nonlocal boundary value problem by applying Leray-Schauder Continuation Principle. To illustrate the applicability of the obtained results, we consider an example.

An Extension of Strict Almost Contractions Employing Control Function and Binary Relation with Applications to Boundary Value Problems

This article comprises some fixed point results for Boyd–Wong-type strict almost contractions using locally L-transitive binary relations. We provide several examples to illustrate our findings. On applying our results, we determine a unique solution of a special boundary value problem.

A superconvergent finite node method for semilinear elliptic problems

This paper proposes and analyzes a superconvergent finite node method (SFPM) for meshless solution of semilinear boundary value problems with variable coefficients. Smoothed derivatives of the moving least squares approximation are adopted in the SFPM to achieve the high accuracy and favorable superconvergence. Theoretical accuracy analysis of the SFPM and the traditional FPM is detailedly presented through local truncation error analysis. Some numerical results are provided to demonstrate the superconvergence and theoretical results of the method.

Boundary Value Problem for a Loaded Pseudoparabolic Equation with a Fractional Caputo Operator

Differential equations containing fractional derivatives, for both time and spatial variables, have now begun to attract the attention of mathematicians and physicists; they are used in connection with these equations as mathematical models of various processes. The fractional derivative equation tool plays a crucial role in describing plenty of natural processes concerning physics, biology, geology, and so on. In this paper, we studied a loaded equation in relation to a spatial variable for a linear pseudoparabolic equation, with an initial and second boundary value condition (the Neumann condition), and a fractional Caputo derivative. A distinctive feature of the considered problem is that the load at the point is in the higher partial derivatives of the solution. The problem is reduced to a loaded equation with a nonlocal boundary value condition. A way to solve the considered problem is by using the method of energy inequalities, so that a priori estimates of solutions for non-local boundary value problems are obtained. To prove that this nonlocal problem is solvable, we used the method of continuation with parameters. The existence and uniqueness theorems for regular solutions are proven.

Positive Solutions for Third-Order Boundary Value Problems with Indefinite Weight

Positive Solutions for Third-Order Boundary Value Problems with Indefinite Weight

On the Numerical Solution of Boundary Value Problem (BVP) of the Ordinary Differential Equation (ODE) - The Case of Steady-State Bio-Heat Equation with Combined Heat Transfer Coefficient by Pseudo-Spectral Collocation Method

Spectral methods for the solution of a boundary value problem of an ordinary differential equation are reviewed with particular emphasis laid on pseudo-spectral collocation method. The pseudo-collocation method is then used to solve the one dimensional bio-heat equation with metabolic heat generation in cylindrical coordinates applied to human tissue. It was noticed that an increase in heat transfer coefficient (hA), enhanced the temperature but a decrease in the tissue thickness was observed when this coefficient was increased. The effects of the combined heat transfer coefficient are analyzed and the results indicate that the obtained solution can be used in the study of the thermal behaviour of a biological system with the potential to locate tumours in the living tissue.

Application of artificial neural networks in atomic force microscopy

The paper describes a method of artificial neural network application to build a computer database for numerical simulation of the process of indentation of an atomic force microscope probe into an elastomeric composite with a granular filler (nonlinear elastic medium with rigid spherical inclusions). The use of such a base makes it possible to significantly improve the speed and quality of interpretation of the results of nanoindentation for structurally inhomogeneous materials. In this case, information becomes available not only about what is done on the surface of the sample but also in the near-surface layer inside it. An algorithm has been developed with the help of which an artificial neural network was built and “trained,” designed to obtain indentation curves depending on the size of the filler particles and its localization in the near-surface layer of the composite (depth and horizontal distance from the top of the AFM probe). It is shown that the speed of constructing indentation curves increases by several orders of magnitude compared to conventional approaches based on the numerical solution of the corresponding boundary value problems for each specific case. Accordingly, computer costs are also significantly reduced, that is, in the presence of an already built and “trained” neural network, powerful and high-speed computers are not needed.