Linear Boundary Value Problems Research Articles

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.

Multilevel correction adaptive finite element method for solving nonsymmetric eigenvalue problems

Large-scale nonsymmetric eigenvalue problems are common in various fields of science and engineering computing. However, their efficient handling is challenging, and research on their solution algorithms is limited. In this study, a new multilevel correction adaptive finite element method is designed for solving nonsymmetric eigenvalue problems based on the adaptive refinement technique and multilevel correction scheme. Different from the classical adaptive finite element method, which requires solving a nonsymmetric eigenvalue problem in each adaptive refinement space, our approach requires solving a symmetric linear boundary value problem in the current refined space and a small-scale nonsymmetric eigenvalue problem in an enriched correction space. Since it is time-consuming to solve a large-scale nonsymmetric eigenvalue problem directly in adaptive spaces, the proposed method can achieve nearly the same efficiency as the classical adaptive algorithm when solving the symmetric linear boundary value problem. In addition, the corresponding convergence and optimal complexity are verified theoretically and demonstrated numerically.

Solvability Criterion for Linear Boundary-Value Problems for Integrodifferential Fredholm Equations with Degenerate Kernels in Banach Spaces

Solvability Criterion for Linear Boundary-Value Problems for Integrodifferential Fredholm Equations with Degenerate Kernels in Banach Spaces

ON A METHOD FOR SOLVING A FAMILY OF NONLINEAR BOUNDARY VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS

In this paper, we consider a boundary value problem for a family of linear differential equations that obey a family of nonlinear two-point boundary conditions. For each fixed value of the family parameter, the boundary value problem under study is a nonlinear two-point boundary value problem for a system of ordinary differential equations. Non-local boundary value problems for systems of partial differential equations, in particular, non-local boundary value problems for systems of hyperbolic equations with mixed derivatives, can be reduced to the family of boundary value problems for ordinary differential equations. Therefore, the establishment of solvability conditions and the development of methods for solving a family of boundary value problems for differential equations are actual problems. In this paper, using the ideas of the parametrization method of D. S. Dzhumabaev, which was originally developed to establish the signs of unambiguous solvability of a linear two-point boundary value problem for a system of ordinary equations, a method for finding a numerical solution to the problem under consideration is proposed.

Boundary-value problems for weakly singular integral equations

<p style='text-indent:20px;'>We consider a perturbed linear boundary-value problem for a weakly singular integral equation. Assume that the generating boundary-value problem is unsolvable for arbitrary inhomogeneities. Efficient conditions for the coefficients guaranteeing the appearance of the family of solutions of the perturbed linear boundary-value problem in the form of Laurent series in powers of a small parameter <inline-formula><tex-math id="M1">\begin{document}$ \varepsilon $\end{document}</tex-math></inline-formula> with singularity at the point <inline-formula><tex-math id="M2">\begin{document}$ \varepsilon = 0 $\end{document}</tex-math></inline-formula> are established.</p>

On successive linearization method for differential equations with nonlinear conditions

Abstract This paper presents a new technique for solving linear and nonlinear boundary value problems subject to linear or nonlinear conditions. The technique is based on the blending of the Chebyshev pseudospectral method. The rapid convergence and effectiveness are verified by several linear and nonlinear examples, and results are compared with the exact solutions. Our results show a remarkable improvement in the convergence of the results when compared with exact solutions.

Theoretical Investigation on Performance Characteristics of Aerostatic Journal Bearings with Active Displacement Compensator

Active aerostatic bearings are capable of providing negative compliance, which can be successfully used to automatically compensate for deformation of the machine tool system in order to reduce the time and improve the quality of metalworking. The article considers an aerostatic radial bearing with external combined throttling systems and an elastic displacement compensator, which is an alternative to aerostatic bearings with air flow rate compensators. The results of the mathematical modeling and theoretical research of stationary and nonstationary modes of operation of bearings with slotted and diaphragm throttling systems are presented. A counter-matrix sweep method has been developed for solving linear and nonlinear boundary value problems in partial derivatives with respect to the function of the square of the pressure in the bearing gap and inter-throttling bearing cavities for any values of the relative shaft eccentricity. A numerical method is proposed for calculating the dynamic quality criteria, and the transfer function of the dynamic compliance of a bearing with small displacements is considered as a linear automatic control system with distributed parameters. An experimental verification of the theoretical characteristics of the bearing was carried out, which showed a satisfactory correspondence among the compared data. It is shown that bearings with a throttle system have the best quantitative and qualitative load characteristics. The possibility of optimal determination of the values of a number of important parameters that provide the bearing with optimal performance and a high stability margin is established. It is shown that bearings with an elastic suspension of the movable sleeve allow one to compensate for significant movements, which can be larger than the size of the air gap by an order of magnitude or more. In these conditions, similar bearings with air flow compensators would be obviously inoperative.

Boundary-Value Problems for Systems of Linear Difference-Algebraic Equations

We establish constructive conditions for the solvability of linear boundary-value problem for a system of difference-algebraic equations and develop a scheme for the construction of solutions of this problem. We also obtain sufficient conditions for the reducibility of a difference-algebraic system to a sequence of systems combining difference and algebraic equations. We propose an original classification and a unified scheme of construction of the solutions of difference-algebraic boundary-value problems.

The effect of local mass displacement on coupled fields in dielectrics

The classical theory is incapable of explaining the size-dependent behavior of small-scale structures. This theory also fails to describe a linear response of electric polarization to the temperature gradient. The mentioned effects can be explained by the local gradient theory of electrothermoelasticity that alongside the mechanical, thermal, and electromagnetic processes considers the effect of the changes in material microstructure on the solid behavior. Within the local gradient theory, the above changes in the material microstructure are linked with the process of the local mass displacement. The basic equations of this theory were obtained earlier based on the continuum-thermodynamic approach. In this study, the variational approach for the formulation of linear boundary-value problems of the local gradient electrothermoelasticity is proposed. Using this approach, the fundamental equations and proper boundary conditions which describe the physical behavior of the linear continuum are derived. The usefulness of the proposed theory is verified based on two simple problems. Namely, the theory is used to study the response of the electric polarization of the isotropic layer to the temperature gradient and to investigate the coupled field in a simply-supported piezoelectric nanobeam subjected to the distributed uniform force. A generalized mathematical model of the Bernoulli–Euler electroelastic beam is formulated. It is shown that the resulting local gradient beam model can capture the size-depending behavior of the piezoelectric beam at the nanoscale. The results obtained in this study are general and can be useful for modeling new devices based on size, thermopolarization, and piezoelectric effects.

On One Algorithm To Find a Solution to a Linear Two-Point Boundary Value Problem

On One Algorithm To Find a Solution to a Linear Two-Point Boundary Value Problem

Mathematical modelling and design of isolation diaphragms for pressure gauges

The paper discusses some questions of equilibrium, stability and shape optimization of circular membranes with arbitrary profile along the meridian. Such membranes can be used as a flexible barrier to isolate the media on one side of a diaphragm from media on the other side, whilst allowing the transmission of pressure with minimal loss. The main goal of the analysis is to propose techniques for designing such shape of corrugated membranes that satisfy the requirements of so call “slow growing linear or almost linear” behavior of the pressure-volume diagram. The applied Kirchhoff-Love theory for non-shallow shells was used as a base for deriving nonlinear as well as linearized boundary value problems of equilibrium. A genetic algorithm was used to evaluate and optimize the characteristics of corrugated membranes. The presented analysis of the membrane with the maximum linear stroke showed its efficiency and reliability. At the same time, special attention should be paid to the basic parametrization used in modelling, since the mathematically optimal solution is not always suitable from the point of view of membrane production technology.

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.

Energy-optimal trajectory problems in relative motion solved via Theory of Functional Connections

In this paper, we present a new approach for solving a broad class of energy-optimal trajectory problems in relative motion using the recently developed Theory of Functional Connections (TFC). A total of four problem cases are considered and solved, i.e. rendezvous and intercept with fixed and free final time. Each problem is constrained and formulated using an indirect approach which casts the optimal trajectory problem as a system of linear or nonlinear two-point boundary value problems for the fixed and free final time cases, respectively. Using TFC, we convert each two-point boundary value problem into an unconstrained problem by analytically embedding the boundary constraints into a “constrained expression.” The latter includes a free-function that is expanded using Chebyshev polynomials with unknown coefficients. Regardless of the values of the unknown coefficients, the boundary constraints are satisfied and simple optimization schemes can be employed to numerically solve the problem (e.g. linear and nonlinear least-square methods). To validate the proposed approach, the TFC solutions are compared with solutions obtained via an analytical based method as well as direct and indirect numerical methods. In general, the proposed technique produces solutions to machine level accuracy. Additionally, for the cases tested, it is reported that computational run-time within the MATLAB implementation is lower than 28 and 300 ms for the fixed and free final time problems respectively. Consequently, the proposed methodology is potentially suitable for on-board generation of optimal trajectories in real-time.

A two-step hybrid block method with fourth derivatives for solving third-order boundary value problems

This manuscript proposes an implicit two-step hybrid block method which incorporates fourth derivatives, for solving linear and non-linear third-order boundary value problems in ODEs. The derivation of the present method is based on collocation and interpolation techniques, and the convergence analysis of the new strategy is proved to be seventh-order convergent. The proposed approach produces discrete approximations at the grid points, obtained after solving an algebraic system of equations. Numerical experiments are studied to show the performance and viability of the proposed approach. The numerical results demonstrated that the new technique gives accurate approximations, which are better than some existing strategies in the available literature and also found to be in good agreement with known analytical solutions.

Boundary Value Problems Numerical Solution on Multiblock Grids

Results of theoretical analysis of the geometric multigrid algorithms convergence are presented for solving the linear boundary value problems on a two-block grid. In this case, initial domain could be represented as a union of intersecting subdomains, in each of them a structured grid could be constructed generating a hierarchy of coarse grids. Multigrid iteration matrix is obtained using the damped nonsymmetric iterative method as a smoother. The multigrid algorithm contains a new problem-dependent component --- correction interpolation between grid blocks. Smoothing property for the damped nonsymmetric iterative method and convergence of the robust multigrid technique are proved. Estimation of the multigrid iteration matrix norm is obtained (sufficient convergence condition). It is shown that the number of multigrid iterations does not depend on either the step or the number of grid blocks, if interpolation of the correction between grid blocks is sufficiently accurate. Results of computational experiments are presented on solving the three-dimensional Dirichlet boundary value problem for the Poisson equation illustrating the theoretical analysis. Results obtained could be easily generalized to multiblock grids. The work is of interest for developers of highly efficient algorithms for solving the (initial-) boundary value problems describing physical and chemical processes in complex geometry domains

A multigrid method for the ground state solution of Bose–Einstein condensates based on Newton iteration

In this paper, a new kind of multigrid method is proposed for the ground state solution of Bose–Einstein condensates based on Newton iteration scheme. Instead of treating eigenvalue \(\lambda \) and eigenvector u separately, we regard the eigenpair \((\lambda , u)\) as one element in the composite space \({\mathbb {R}} \times H_0^1(\varOmega )\) and then Newton iteration step is adopted for the nonlinear problem. Thus in this multigrid scheme, the main computation is to solve a linear discrete boundary value problem in every refined space, which can improve the overall efficiency for the simulation of Bose–Einstein condensations.

A Legendre reproducing kernel method with higher convergence order for a class of singular two-point boundary value problems

We study a higher-order Legendre reproducing kernel method (LRKM) for singular two-point boundary value problems (BVPs). Our method relies upon Legendre polynomials. We carry out error estimatation by using interpolation theory. Various numerical tests including linear and nonlinear problems are performed to reveal the stability and efficiency of the new scheme. Therewith, as an application of the proposed approach, a famous time-dependent partial differential equation, namely Allen–Cahn test problem, is transfomed into the associated system of linear BVPs by localizing in temporal variable and is challenged numerically in details by the LRKM.

Reformulated Adomian Decomposition Method for the Approximation of Special Linear and Nonlinear Two-Point Boundary Value Problems

Boundary Value Problems (BVPs) of higher order have been found to be potentially applicable in hydro-magnetic stability, hydrodynamics, chemical reactions, heat power transmission theory and the boundary layer theory in fluid mechanics. In this research, a method which decomposes the solution into the series which converges rapidly shall be derived. We shall call this method the Reformulated Adomian Decomposition Method (RADM). This method is an improvement over Adomian Decomposition Method (ADM). The RADM is derived in such a way that on imposing the boundary conditions on the approximant, a system of equations is obtained which in turn is solved for the undetermined constants. On substituting the resulting constants into the solution function we obtain a series solution to the problem. The RADM is applied on some linear and nonlinear two-point BVPs and from the results obtained, the method is said to be computationally reliable.

Venttsel Boundary Value Problems with Discontinuous Data

We study linear and quasilinear Venttsel boundary value problems involving elliptic operators with discontinuous coefficients. On the basis of the a priori estimates obtained, maximal regularity and strong solvability in Sobolev spaces are proved.

Numerical Solution of the Seventh Order Boundary Value Problems Using B-Spline Method

We develop a numerical method for solving the boundary value problem of The Linear Seventh Ordinary Boundary Value Problem by using seventh degree B-Spline function. Formulation is based on particular terms of order of seventh order boundary value problem. We obtain Septic B-Spline formulation and the Collocation B-spline Method is formulated as an approximation solution. We apply the presented method to solve an example of seventh-order boundary value problem which the results show that there is an agreement between approximate solutions and exact solutions. Resulting low absolute errors show that the presented numerical method is effective for solving high order boundary value problems. Finally, a general conclusion has been included.