Wave scattering by a permeable barrier over undulating bed topography

In this work, a technique based on combination between collocation and spline method along with the shooting method is proposed for the solution of boundary value problem of order 2. Numerical results show that the method is simple and effective.

There are few numerical techniques available to solve third order boundary value problems. In this paper, we present Galerkin weighted residual method for constructing the numerical solution of third order linear and nonlinear boundary value problems with two point boundary conditions. The method is formulated as a rigorous matrix form. Several numerical examples of both linear and nonlinear boundary value problems in the literature are presented to illustrate the reliability and efficiency of the proposed method. The present method is quite efficient and yields better results when compared with the existing methods in the literature. Keywords: Galerkin method, Third order linear and nonlinear BVPs, Bernstein Polynomials.

Background/Objectives: Numerically solving second order Boundary Value Problems (BVPs) directly is an area that has drawn much attention in recent literature and hence the general aim of this article. Methods/Statistical Analysis: Block methods are developed using Taylor series approach and are applied to directly approximate the considered second order Boundary Value Problems. Certain specifications of the block method developed involves it being of step number k = 4 and order p = 5. This block method consists of mathematical expressions that will simultaneously provide results at the grid points and derivatives of the grid points for the k-step method. Findings: After the development of the new block method in this article, certain Boundary Value Problems are considered and the new block method is adopted to solve it numerically and comparison was made with their exact solution. The results show the new block method show superiority in comparison to other numerical methods of same order p = 5 adopted by other authors in previous literature. Hence, this new block method can be considered a more suitable numerical approach to approximate the numerical solutions of second order BVPs. Applications/Improvements: This block method is suitable for application in solving numerically second order BVPs with closer accuracy to the exact or analytic solutions of the differential equations under consideration.

Solution of a nonlinear boundary value problem for a second-order ordinary differential equation by the method of finite differences

Numerical solution of two-point boundary value problems

We study the regularity of the solutions of second order boundary value problems on manifolds with boundary and bounded geometry. We first show that the regularity property of a given boundary value problem $(P, C)$ is equivalent to the uniform regularity of the natural family $(P_x, C_x)$ of associated boundary value problems in local coordinates. We verify that this property is satisfied for the Dirichlet boundary conditions and strongly elliptic operators via a compactness argument. We then introduce a uniform Shapiro-Lopatinski regularity condition, which is a modification of the classical one, and we prove that it characterizes the boundary value problems that satisfy the usual regularity property. We also show that the natural Robin boundary conditions always satisfy the uniform Shapiro-Lopatinski regularity condition, provided that our operator satisfies the strong Legendre condition. This is achieved by proving that "well-posedness implies regularity" via a modification of the classical "Nirenberg trick". When combining our regularity results with the Poincare inequality of (Ammann-Grosse-Nistor, preprint 2015), one obtains the usual well-posedness results for the classical boundary value problems in the usual scale of Sobolev spaces, thus extending these important, well-known theorems from smooth, bounded domains, to manifolds with boundary and bounded geometry. As we show in several examples, these results do not hold true anymore if one drops the bounded geometry assumption. We also introduce a uniform Agmon condition and show that it is equivalent to the coerciveness. Consequently, we prove a well-posedness result for parabolic equations whose elliptic generator satisfies the uniform Agmon condition.

The main goal of this study is to find the solution of initial boundary value problem for the one-dimensional time and space-fractional diffusion equation which is a very intriguing topic for many researchers. With the aim of newly defined inner product, which is the main contribution of this study, the analytic solution of the boundary value problem is obtained. The time and space-fractional derivatives are defined in the Caputo sense which is more suitable than Riemann-Liouville sense. We apply the separation of variables method to reduce the problem to two separate fractional ODEs. The generalized solution is constructed/formed in the form of a Fourier series with respect to the eigenfunctions of a certain eigenvalue problem. In order to obtain the coefficients of the Fourier series for the solution, we define a new inner product which is the key point of study.

In this paper, we consider the existence of solutions for the discrete mixed boundary value problems involvingp,q-Laplacian operator. By using critical points theory, we obtain the existence of at least two positive solutions for the boundary value problem under appropriate assumptions on the nonlinearity.

The standard three-point discretization applied to the numerical solution of nonlinear boundary value problems for second-order systems with a singularity of the first kind is investigated. The results are extended to the boundary value problems arising in practical problems from mechanics and chemistry. A number of numerical examples illustrating the theoretical results is presented.

The boundary element method (BEM) is an efficient numerical method that approximates solutions of various boundary value problems. Despite its success little research has been performed on the conditioning of the linear systems that appear in the BEM. For a Laplace equation with Dirichlet boundary conditions a remarkable phenomenon is observed; the corresponding boundary integral equation (BIE) is singular for a certain critical size of the 2D domain. As a consequence the discrete counterpart of the BIE, the linear system, is singular too, or at least ill-conditioned. This is reflected by the condition number of the system matrix, which is infinitely large, or at least very large. When the condition number of the BEM-matrix is large, the linear system is difficult to solve and the solution of the system is very sensible to perturbations in the boundary data. For a Laplace equation with mixed boundary conditions a similar phenomenon is observed. The corresponding BEM-matrix consists of two blocks; one block originates from the BEM-matrix belonging to the Dirichlet problem, the other block originates from the BEM-matrix belonging to the Neumann problem. The composite matrix inherits the solvability problems from the Dirichlet block. In other words, for the Laplace equation with mixed boundary conditions there exists also a critical size of the 2D domain for which the BEM-matrix has an infinitely large condition number. Hence the size and shape of the domain affects the solvability of the BEM problem. The critical size of the domain for which the BIE becomes singular is related to the logarithmic capacity of the domain. The logarithmic capacity is a positive real number that is a function of the size and shape of the domain. If this logarithmic capacity is equal to one, the domain is a critical domain, and for this domain the BIE becomes singular. Thus by computing the logarithmic capacity we can a-priori determine whether the BIE will be singular or not. The logarithmic capacity depends linearly on the scale of the domain, and thus a domain with logarithmic capacity equal to one can always be found by rescaling the domain. Unfortunately the logarithmic capacity can only be computed analytically for a few simple domains; for more involved domains the logarithmic capacity can be estimated though. There are several possibilities to avoid large condition numbers, i.e. singular BIEs that appear at critical domains. The first option is to rescale the domain such that the logarithmic capacity is unequal to one. One can also add a supplementary condition to the BIE and the linear system. A drawback of this option is that the linear system has more equations than unknowns and different techniques are required to solve the system. A third option is to slightly modify the fundamental solution of the Laplace operator. This fundamental solution directly appears in the BIE and it can be shown that a suitable modification yields BIEs that do not become singular. The critical domains for which the BIEs become singular do not restrict to Laplace equations only. Also for BIEs applied to the biharmonic equation or the elastostatic equations and the Stokes equations such critical domains exist. As the last two equations are vectorial equations, also the corresponding BIE consists of two equations. As a consequence two critical domains can be found for which these BIEs become singular. To obtain nonsingular BIEs techniques similar to the Laplace case can be used. Unfortunately we cannot a-priori determine the sizes for which the BIEs becomes singular, and thus do not know to what size we should rescale the domain to obtain nonsingular BIEs. The existence of critical domains is in essence caused by the logarithmic term in the fundamental solutions for the elliptic boundary value problems in 2D. This logarithmic term does not depend linearly on the size of the domain. When a domain is scaled, i.e. multiplied by a scale factor, the argument of the logarithm is also multiplied by this scale factor, but the logarithm turns this into an additive term. Thus the logarithm transforms multiplication into addition. This affects the BIEs in such a way that critical domains can appear. The fundamental solutions of boundary value problems in 3D do not contain a logarithmic term. Hence scaling of the domain does not affect the fundamental solution, and consequently also the BIE is not affected. Hence we may safely rescale 3D domains without the risk to encounter a critical domain. An example in which a domain takes many different sizes and shapes is the blowing problem. In this problem a viscous fluid is blown to a desired shape. Typically the time is discretised into a set of discrete time steps, and at each step the shape of the fluid is computed by solving the Stokes equations. When attempting to simulate this problem in 2D, we meet a large number of 2D domains, and we risk that one of these domains is equal to or approaches a critical domain. In such a case the BEM will have difficulties with solving the Stokes equations for that particular domain. When simulating the blowing problem in 3D, no critical domains are encountered. It turns out that the BEM is a very efficient numerical method for this particular 3D problem with a moving boundary. As we are merely interested in the shape of the fluid, we only need to know the flow of its boundary. The BEM does exactly that; it does not compute the flow at the interior of the fluid. Furthermore it is rather easy to include other effects from the blowing problem in the model, such as gravity, surface tension and friction from the contact of the fluid with a wall. As only the boundary of the fluid is discretised, the system matrices that appear in the BEM are smaller than the system matrices that appear when solving the problem with a finite element method, for example. Though the BEM-matrices are dense, while the finite element matrices are sparse, the computational effort for the BEM is relatively low. In short, the BEMis a very appropriate numerical method when solving blowing problems.

By the notation and monotone convergence theorem of Henstock-Kurzweil integral, we investigate the existence of continuous solutions for the second order boundary value problems with integral boundary conditions in which the nonlinearities are allowed to have the singularities in t and are not Lebesgue integrable.

Symmetric positive solutions of boundary value problems with integral boundary conditions

We present two collocation algorithms based on interpolating and approximating subdivision schemes for the solution of fourth order boundary value problems arising in the mathematical modeling of viscoelastic, and inelastic flows, deformation of beams, arches, and load bearing members like street lights and robotic arms in multipurpose engineering systems. Numerical examples are given to illustrate the algorithms. We conclude that approximating schemes based collocation algorithms give better solution than interpolating schemes based collocation algorithms. Main purpose of this paper is to explore and seek the applications of interpolating and approximating subdivision schemes in the field of boundary value problems along with intrinsic comparison of the results obtained by algorithms based on these schemes. A comparison with other approaches of this type of boundary value problems in order to see the advantages of the proposed methods is also given.

In this paper, we mainly discussed an inhomogeneous Riemann-Hilbert boundary value problem for complex partial differential operators of higher order on the bicylinder $ D^2 $ in $ \mathbb{C}^2 $. Applying the classical Cauchy-Pompeiu formula and the Gauss theorem, we found out the solvable condition and obtained the specific solution of the inhomogeneous boundary value problem on the bicylinder $ D^2 $. The conclusion lays a solid foundation for further research on other types of boundary value problems concerning complex partial differential equations, such as mixed boundary value problems.

Usually, the boundary value problems of the theory of elasticity are formulated with respect to displacements, and are reduced to the well-known Lame equations. Strains and stresses can be calculated from displacements as a solution to Lame's equation. Also known are the Beltrami Mitchell equations, which make it possible to formulate the boundary value problem of the theory of elasticity with respect to stresses. Currently, the boundary value problems of the theory of elasticity in stresses are studied in more detail in the two-dimensional case, and usually solved numerically with the introduction of the Airy stress function. But, the direct solution of boundary value problems of elasticity theory with respect to stresses requires further researches. This work, similarly to the boundary value problem in stresses, is devoted to the formulation and numerical solution of boundary value problems of the theory of elasticity with respect to deformations. The proposed boundary value problem consists of six Beltrami-Mitchell-type equations depending on strains and three equations of the equilibrium equation expressed with respect to deformations. As boundary conditions, in addition to the usual conditions for surface forces, three additional conditions are also introduced based on the equilibrium equations. The boundary value problem is considered in detail for a rectangular area. The discrete analogue of the boundary value problem is composed by the finite difference method. The convergence of difference schemes and an iterative method for their solution are studied. Software has been developed in the C++ environment for solving boundary value problems in the theory of elasticity and deformation. A number of boundary value problems on the deformation of a rectangular plate are solved numerically under various boundary conditions. The reliability of the obtained results is substantiated by comparing the numerical results, with the exact solution, as well as with the known solutions of the plate tension problems with parabolic and uniformly distributed edge loads.