PDE Problems Research Articles

Best Approximation in Hardy Spaces and by Polynomials, with Norm Constraints

Two related approximation problems are formulated and solved in Hardy spaces of the disc and annulus. With practical applications in mind, truncated versions of these problems are analysed, where the solutions are chosen to lie in finite-dimensional spaces of polynomials or rational functions, and are expressed in terms of truncated Toeplitz operators. The results are illustrated by numerical examples. The work has applications in systems identification and in inverse problems for PDEs.

A Simple Method to Price Window Reset Options

A window reset option is a kind of reset options with continuous reset constraints. The issue is very important for applying to employee stock options in finance or reservation options on truck-only toll lanes in traffic management. Our contribution of this study is that we proposed an accurate and simple method to price window reset options. The option price is formulated as the solution of a boundary value problem of the Black-Scholes PDE. The problem is then transformed into an initial-boundary value problem of the heat equation. Then Green’s function is applied to solve the heat equation problem. Finally, the option price is calculated numerically. A numerical example and some discussions are presented in this paper.

Dividends Sharing Convertible Bonds Pricing and Numerical Evaluation

The convertible bond is becoming one of the most important financial instruments for the company to raise capital fund since it was first issued by American New York Erie Company in 1843. In this paper, it is the first time to study the pricing problem for convertible bond whose underlying stocks pay dividends via the reflected backward stochastic differential equations. Associating the solutions of reflected BSDEs with the obstacle problems for nonlinear parabolic PDEs, we establish the pricing formulas for convertible bonds with continuous and discrete dividends by means of the viscosity solutions for some PDEs. Besides, we also derive the price of convertible bonds with higher borrowing rate which is realistic in the financial market. Then the numerical evaluations are provided by the radial basis functions method. Moreover, we discuss the influence of dividends paying as well as higher borrowing rate on the convertible bond price at last.

Inverse problems for DEs and PDEs using the collage theorem: a survey

In this paper, we present several methods based on the collage theorem and its extensions for solving inverse problems for initial value and boundary value problems. Several numerical examples show the quality of this approach and its stability. At the end we present an application to the Euler-Bernoulli beam equation with boundary measurements.

The finite volume method for solving systems of non-linear initial-boundary value problems for PDE's

The finite volume method for solving systems of non-linear initial-boundary value problems for PDE's

Well-Posed and Ill-Posed Boundary Value Problems for PDE 2013

1 Department of Mathematics, Fatih University, Buyukcekmece, 34500 Istanbul, Turkey 2 Science Research Computer Center of Moscow State University, Vorobjevy Gory, Moscow 119899, Russia 3 Department of Mathematics and Statistics, College of Science, Sultan Qaboos University, 123 Muscat, Oman 4 Universiti Putra Malaysia, Institute of Advanced Technology, 43400 Serdang, Selangor, Malaysia 5 American Islamic College, Chicago, IL 60613, USA

Harmonic averages, exact difference schemes and local Green’s functions in variable coefficient PDE problems

AbstractA brief survey is given to show that harmonic averages enter in a natural way in the numerical solution of various variable coefficient problems, such as in elliptic and transport equations, also of singular perturbation types. Local Green’s functions used as test functions in the Petrov-Galerkin finite element method combined with harmonic averages can be very efficient and are related to exact difference schemes.

Stable Computation of Differentiation Matrices and Scattered Node Stencils Based on Gaussian Radial Basis Functions

Radial basis function (RBF) approximation has the potential to provide spectrally accurate function approximations for data given at scattered node locations. For smooth solutions, the best accuracy for a given number of node points is typically achieved when the basis functions are scaled to be nearly flat. This also results in nearly linearly dependent basis functions and severe ill-conditioning of the interpolation matrices. Fornberg, Larsson, and Flyer recently generalized the RBF-QR method to provide a numerically stable approach to interpolation with flat and nearly flat Gaussian RBFs for arbitrary node sets in up to three dimensions. In this work, we consider how to extend this method to the task of computing differentiation matrices and stencil weights in order to solve partial differential equations. The expressions for first and second order derivative operators as well as hyperviscosity operators are established, numerical issues such as how to deal with non-unisolvency are resolved, and the accuracy and computational efficiency of the method are tested numerically. The results indicate that using the RBF-QR approach for solving PDE problems can be very competitive compared with using the ill-conditioned direct solution approach or using variable precision arithmetic to overcome the conditioning issue.

Monge–Ampére based moving mesh methods for numerical weather prediction, with applications to the Eady problem

We derive a moving mesh method based upon ideas from optimal transport theory which is suited to solving PDE problems in meteorology. In particular we show how the Parabolic Monge–Ampére method for constructing a moving mesh in two-dimensions can be coupled successfully to a pressure correction method for the solution of incompressible flows with significant convection and subject to Coriolis forces. This method can be used to resolve evolving small scale features in the flow. In this paper the method is then applied to the computation of the solution to the Eady problem which is observed to develop large gradients in a finite time. The moving mesh method is shown to work and be stable, and to give significantly better resolution of the evolving singularity than a fixed, uniform mesh.

Fully pseudospectral time evolution and its application to [formula omitted] dimensional physical problems

It was recently demonstrated that time-dependent PDE problems can numerically be solved with a fully pseudospectral scheme, i.e. using spectral expansions with respect to both spatial and time directions [15]. This was done with the example of simple scalar wave equations in Minkowski spacetime. Here we show that the method can be used to study interesting physical problems that are described by systems of nonlinear PDEs. To this end we consider two 1+1 dimensional problems: radial oscillations of spherically symmetric Newtonian stars and time evolution of Gowdy spacetimes as particular cosmological models in general relativity.

Potential evaluation in space–time BIE formulations of 2D wave equation problems

In this paper we examine the computation of the potential generated by space–time BIE representations associated with Dirichlet and Neumann problems for the 2D wave equation. In particular, we consider the efficient evaluation of the (convolution) time integral that appears in the potential representation. For this, we propose two simple quadrature rules which appear more efficient than the currently used ones. Both are of Gaussian type: the classical Gauss–Jacobi quadrature rule in the case of a Dirichlet problem, and the classical Gauss–Radau quadrature rule in the case of a Neumann problem. Both of them give very accurate results by using a few quadrature nodes, as long as the potential is evaluated at a point not very close to the boundary of the PDE problem domain. To deal with this latter case, we propose an alternative rule, which is defined by a proper combination of a Gauss–Legendre formula with one of product type, the latter having only 5 nodes.The proposed quadrature formulas are compared with the second order BDF Lubich convolution quadrature formula and with two higher order Lubich formulas of Runge–Kutta type. An extensive numerical testing is presented. This shows that the new proposed approach is very competitive, from both the accuracy and efficiency points of view.

Towards practical anisotropic adaptive FEM on triangular meshes: a new refinement paradigm

Adaptive anisotropic refinement of finite element meshes allows one to reduce the computational effort required to achieve a specified accuracy of the solution of a PDE problem. We present a new approach to adaptive refinement and demonstrate that this allows one to construct algorithms which generate very flexible and efficient anisotropically refined meshes, even improving the convergence order compared to adaptive isotropic refinement if the problem permits.

Numerics of boundary-domain integral and integro-differential equations for BVP with variable coefficient in 3D

A numerical implementation of the direct boundary-domain integral and integro-differential equations, BDIDEs for treatment of the Dirichlet problem for a scalar elliptic PDE with variable coefficient in a three-dimensional domain is discussed. The mesh-based discretisation of the BDIEs with tetrahedron domain elements in conjunction with collocation method leads to a system of linear algebraic equations (discretised BDIE). The involved fully populated matrices are approximated by means of the H-Matrix/adaptive cross approximation technique. Convergence of the method is investigated.

Blind backscattering experimental data collected in the field and an approximately globally convergent inverse algorithm

An approximately globally convergent numerical method for a 1D coefficient inverse problem for a hyperbolic PDE is applied to image dielectric constants of targets from blind experimental data. The data were collected in the field by the Forward Looking Radar of the US Army Research Laboratory. A posteriori analysis has revealed that computed and tabulated values of dielectric constants are in good agreement. Convergence analysis is presented.

Approximating the dynamics of communicating cells in a diffusive medium by ODEs—homogenization with localization

Bacteria may change their behavior depending on the population density. Here we study a dynamical model in which cells of radius [R] within a diffusive medium communicate with each other via diffusion of a signalling substance produced by the cells. The model consists of an initial boundary value problem for a parabolic PDE describing the exterior concentration [u] of the signalling substance, coupled with [N] ODEs for the masses [ai] of the substance within each cell. We show that for small [R] the model can be approximated by a hierarchy of models, namely first a system of [N] coupled delay ODEs, and in a second step by [N] coupled ODEs. We give some illustrations of the dynamics of the approximate model.

Fold completeness of a system of root vectors of a system of unbounded polynomial operator pencils in Banach spaces. II. Application

This paper is a continuation of the author’s paper in 2009, where the abstract theory of fold completeness in Banach spaces has been presented. Using obtained there abstract results, we consider now very general boundary value problems for ODEs and PDEs which polynomially depend on the spectral parameter in both the equation and the boundary conditions. Moreover, equations and boundary conditions may contain abstract operators as well. So, we deal, generally, with integro-differential equations, functional-differential equations, nonlocal boundary conditions, multipoint boundary conditions, integro-differential boundary conditions. We prove n-fold completeness of a system of root functions of considered problems in the corresponding direct sum of Sobolev spaces in the Banach L q -framework, in contrast to previously known results in the Hilbert L 2-framework. Some concrete mechanical problems are also presented.

A fully non-linear PDE problem from pricing CDS with counterparty risk

In this study, we establish a financial credit derivative pricingmodel for a contract which is subject to counterparty risks. Themodel leads to a fully nonlinear partial differential equationproblem. We study this PDE problem and obtained a solution as thelimit of a sequence of semi-linear PDE problems which also arisefrom financial models. Moreover, the problems and methods build abridge between two main risk frameworks: structure and intensitymodels. We obtain the uniqueness, regularities and some propertiesof the solution of this problem.

The unified method: I. Nonlinearizable problems on the half-line

Boundary value problems for integrable nonlinear evolution PDEs formulated on the half-line can be analyzed by the unified method introduced by one of the authors and used extensively in the literature. The implementation of this general method to this particular class of problems yields the solution in terms of the unique solution of a matrix Riemann–Hilbert problem formulated in the complex k-plane (the Fourier plane), which has a jump matrix with explicit (x, t)-dependence involving four scalar functions of k, called the spectral functions. Two of these functions depend on the initial data, whereas the other two depend on all boundary values. The most difficult step of the new method is the characterization of the latter two spectral functions in terms of the given initial and boundary data, i.e. the elimination of the unknown boundary values. For certain boundary conditions, called linearizable, this can be achieved simply using algebraic manipulations. Here, we present an effective characterization of the spectral functions in terms of the given initial and boundary data for the general case of non-linearizable boundary conditions. This characterization is based on the analysis of the so-called global relation, on the analysis of the equations obtained from the global relation via certain transformations leaving the dispersion relation of the associated linearized PDE invariant and on the computation of the large k asymptotics of the eigenfunctions defining the relevant spectral functions.

The unified method: II. NLS on the half-line with t-periodic boundary conditions

Boundary value problems for integrable nonlinear evolution PDEs formulated on the half-line can be analyzed by the unified method introduced by one of the authors and used extensively in the literature. The implementation of this general method to this particular class of problems yields the solution in terms of the unique solution of a matrix Riemann–Hilbert problem formulated in the complex k-plane (the Fourier plane), which has a jump matrix with explicit (x, t)-dependence involving four scalar functions of k, called spectral functions. Two of these functions depend on the initial data, whereas the other two depend on all boundary values. The most difficult step of the new method is the characterization of the latter two spectral functions in terms of the given initial and boundary data, i.e. the elimination of the unknown boundary values. For certain boundary conditions, called linearizable, this can be achieved by simply using algebraic manipulations. Here, we first present an effective characterization of the spectral functions in terms of the given initial and boundary data for the general case of non-linearizable boundary conditions. This characterization is based on the analysis of the so-called global relation and on the introduction of the so-called Gelfand–Levitan–Marchenko representations of the eigenfunctions defining the spectral functions. We then concentrate on the physically significant case of t-periodic Dirichlet boundary data. After presenting certain heuristic arguments which suggest that the Neumann boundary values become periodic as t → ∞, we show that for the case of the NLS with a sine-wave as Dirichlet data, the asymptotics of the Neumann boundary values can be computed explicitly at least up to third order in a perturbative expansion and indeed at least up to this order are asymptotically periodic.

The unified method: III. Nonlinearizable problems on the interval

Boundary value problems for integrable nonlinear evolution PDEs formulated on the finite interval can be analyzed by the unified method introduced by one of the authors and extensively used in the literature. The implementation of this general method to this particular class of problems yields the solution in terms of the unique solution of a matrix Riemann–Hilbert problem formulated in the complex k-plane (the Fourier plane), which has a jump matrix with explicit (x, t)-dependence involving six scalar functions of k, called the spectral functions. Two of these functions depend on the initial data, whereas the other four depend on all boundary values. The most difficult step of the new method is the characterization of the latter four spectral functions in terms of the given initial and boundary data, i.e. the elimination of the unknown boundary values. Here, we present an effective characterization of the spectral functions in terms of the given initial and boundary data. We present two different characterizations of this problem. One is based on the analysis of the so-called global relation, on the analysis of the equations obtained from the global relation via certain transformations leaving the dispersion relation of the associated linearized PDE invariant and on the computation of the large k asymptotics of the eigenfunctions defining the relevant spectral functions. The other is based on the analysis of the global relation and on the introduction of the so-called Gelfand–Levitan–Marchenko representations of the eigenfunctions defining the relevant spectral functions. We also show that these two different characterizations are equivalent and that in the limit when the length of the interval tends to infinity, the relevant formulas reduce to the analogous formulas obtained recently for the case of boundary value problems formulated on the half-line.