A boundary element method for the Dirichlet eigenvalue problem of the Laplace operator

The solution of eigenvalue problems for partial differential operators by using boundary integral equation methods usually involves some Newton potentials which may be resolved by using a multiple reciprocity approach. Here we propose an alternative approach which is in some sense equivalent to the above. Instead of a linear eigenvalue problem for the partial differential operator we consider a nonlinear eigenvalue problem for an associated boundary integral operator. This nonlinear eigenvalue problem can be solved by using some appropriate iterative scheme, here we will consider a Newton scheme. We will discuss the convergence and the boundary element discretization of this algorithm, and give some numerical results.     

A porous thermoviscoelastic mixture problem: numerical analysis and computational experiments

In this paper, we study, from the numerical point of view, a porous thermoviscoelastic mixture problem. The mechanical problem is written as a linear coupled system of two hyperbolic partial differential equations for the porosities and a parabolic partial differential equation for the temperature field. An existence and uniqueness result and an energy decay property are stated. Then, fully discrete approximations are introduced by using the finite element method to approximate the spatial variable and the backward Euler scheme to discretize the time derivatives. A priori error estimates are proved from which, under suitable regularity conditions, the linear convergence of the algorithm is derived. Finally, some numerical simulations are presented to demonstrate the accuracy of the approximations in an academical one-dimensional example and the behaviour of the solutions in one- and two-dimensional problems.     

Discretized Tikhonov–Phillips regularization for a naturally linearized parameter identification problem

The problem of identification of the diffusion coefficient in the partial differential equation is considered. We discuss a natural linearization of this problem and application of discretized Tikhonov–Phillips regularization to its linear version. Using recent results of regularization theory, we propose a strategy for the choice of regularization and discretization parameters which automatically adapts to unknown smoothness of the coefficient. The estimation of the accuracy will be given and various numerical test supporting theoretical results will be presented.     

Group invariant solution for a pre-existing fracture driven by a power-law fluid in impermeable rock

The effect of power-law rheology on hydraulic fracturing is investigated. The evolution of a two-dimensional fracture with non-zero initial length and driven by a power-law fluid is analyzed. Only fluid injection into the fracture is considered. The surrounding rock mass is impermeable. With the aid of lubrication theory and the PKN approximation a partial differential equation for the fracture half-width is derived. Using a linear combination of the Lie-point symmetry generators of the partial differential equation, the group invariant solution is obtained and the problem is reduced to a boundary value problem for an ordinary differential equation. Exact analytical solutions are derived for hydraulic fractures with constant volume and with constant propagation speed. The asymptotic solution near the fracture tip is found. The numerical solution for general working conditions is obtained by transforming the boundary value problem to a pair of initial value problems. Throughout the paper, hydraulic fracturing with shear thinning, Newtonian and shear thickening fluids are compared.     

Three-layer finite-difference method for linear partial differential-algebraic systems

We consider a boundary value problem for a linear partial differential-algebraic system with a special structure of the matrix pencil, which permits one to split the system by an appropriate transformation into a system of ordinary differential equations, a hyperbolic system, and a linear algebraic system. For the numerical solution of such problems, we use a three-layer method. We prove the theorem on the stability and convergence of the suggested numerical method. The results of numerical experiments are presented as well.     

A superconvergent linear FE approximation for the solution of an elliptic system of PDEs

The aim of this work is to study a nonstandard piecewise linear finite element method for elliptic systems of partial differential equations. This nonstandard method was considered by the authors for scalar elliptic equations and for a planar elasticity problem. The method enables us to compute a superconvergent numerical approximation to the solution of the system of partial differential equations.     

Numerical Solution for the Non-linear Dirichlet Problem of a Branching Process

We give a probabilistic numerical approach for the nonlinear Dirichlet problem associated with a branching process. Main tools are the probabilistic representation of the solution with the measure-valued branching process, as well as specific techniques for the numerical solution of linear partial differential equations, introduced and developed by Milstein and Tretyakov, and Monte Carlo methods.     

Numerical method of solution to loaded nonlocal boundary value problems for ordinary differential equations

A numerical method is suggested for solving systems of nonautonomous loaded linear ordinary differential equations with nonseparated multipoint and integral conditions. The method is based on the convolution of integral conditions into local ones. As a result, the original problem is reduced to an initial value (Cauchy) problem for systems of ordinary differential equations and linear algebraic equations. The approach proposed is used in combination with the linearization method to solve systems of loaded nonlinear ordinary differential equations with nonlocal conditions. An example of a loaded parabolic equation with nonlocal initial and boundary conditions is used to show that the approach can be applied to partial differential equations. Numerous numerical experiments on test problems were performed with the use of the numerical formulas and schemes proposed.     

Three-layer finite difference method for solving linear differential algebraic systems of partial differential equations

A boundary value problem is examined for a linear differential algebraic system of partial differential equations with a special structure of the associate matrix pencil. The use of an appropriate transformation makes it possible to split such a system into a system of ordinary differential equations, a hyperbolic system, and a linear algebraic system. A three-layer finite difference method is applied to solve the resulting problem numerically. A theorem on the stability and the convergence of this method is proved, and some numerical results are presented.     

Numerical solution of the momentum and heat transfer equations for a hydromagnetic flow due to a stretching sheet of a non-uniform property micropolar liquid

A study of the hydromagnetic flow due to a stretching sheet and heat transfer in an incompressible micropolar liquid is made. Temperature-dependent thermal conductivity and a non-uniform heat source/sink render the problem analytically intractable and hence a numerical study is made using the shooting method based on Runge-Kutta and Newton-Raphson methods. The two problems of horizontal and vertical stretching are considered to implement the numerical method. The former problem involves one-way coupling between linear momentum and heat transport equations and the latter involves two-way coupling. Further, both the problems involve two-way coupling between the non-linear equations of conservation of linear and angular momentums. A similarity transformation arrived at for the problem using the Lie group method facilitates the reduction of coupled, non-linear partial differential equations into coupled, non-linear ordinary differential equations. The algorithm for solving the resulting coupled, two-point, non-linear boundary value problem is presented in great detail in the paper. Extensive computation on velocity and temperature profiles is presented for a wide range of values of the parameters, for prescribed surface temperature (PST) and prescribed heat flux (PHF) boundary conditions.     

Numerical solution to the deflection of thin plates using the two-dimensional Berger equation with a meshless method based on multiple-scale Pascal polynomials

In this study, we employ Pascal polynomial basis in the two-dimensional Berger equation, which is a fourth order partial differential equation with applications to thin elastic plates. The polynomial approximation method based on Pascal polynomial basis can be readily adapted to obtain the numerical solutions of partial differential equations. However, a drawback with the polynomial basis is that the resulting coefficient matrix for the problem considered may be ill-conditioned. Due to this ill-conditioned behavior, we use a multiple-scale Pascal polynomial method for the Berger equation. The ill-conditioned numbers can be mitigated using this approach. Multiple scales are established automatically by selecting the collocation points in the multiple-scale Pascal polynomial method. This method is also a meshless method because there is no requirement to establish complex grids or for numerical integration. We present the solutions of six linear and nonlinear benchmark problems obtained with the proposed method on complexly shaped domains. The results obtained demonstrate the accuracy and effectiveness of the proposed method, as well showing its stability against large noise effects.     

Numerical Analysis of a Problem Involving a Viscoelastic Body with Double Porosity

We study from a numerical point of view a multidimensional problem involving a viscoelastic body with two porous structures. The mechanical problem leads to a linear system of three coupled hyperbolic partial differential equations. Its corresponding variational formulation gives rise to three coupled parabolic linear equations. An existence and uniqueness result, and an energy decay property, are recalled. Then, fully discrete approximations are introduced using the finite element method and the implicit Euler scheme. A discrete stability property and a priori error estimates are proved, from which the linear convergence of the algorithm is derived under suitable additional regularity conditions. Finally, some numerical simulations are performed in one and two dimensions to show the accuracy of the approximation and the behaviour of the solution.     

Galerkin approximations with embedded boundary conditions for retarded delay differential equations

Finite-dimensional approximations are developed for retarded delay differential equations (DDEs). The DDE system is equivalently posed as an initial-boundary value problem consisting of hyperbolic partial differential equations (PDEs). By exploiting the equivalence of partial derivatives in space and time, we develop a new PDE representation for the DDEs that is devoid of boundary conditions. The resulting boundary condition-free PDEs are discretized using the Galerkin method with Legendre polynomials as the basis functions, whereupon we obtain a system of ordinary differential equations (ODEs) that is a finite-dimensional approximation of the original DDE system. We present several numerical examples comparing the solution obtained using the approximate ODEs to the direct numerical simulation of the original non-linear DDEs. Stability charts developed using our method are compared to existing results for linear DDEs. The presented results clearly demonstrate that the equivalent boundary condition-free PDE formulation accurately captures the dynamic behaviour of the original DDE system and facilitates the application of control theory developed for systems governed by ODEs.     

Optimal Thickness of a Cylindrical Shell Under a Time-Dependent Force

We discuss thickness optimization problems for cylindrical tubes that are loaded by time-dependent applied force. This is a problem of shape optimization that leads to optimal control in linear elasticity theory. We determine the optimal thickness of a cylindrical tube by minimizing the deformation of the tube under the influence of an external force. The main difficulty is that the state equation is a hyperbolic partial differential equation of the fourth order. The first order necessary conditions for the optimal solution are derived. Based on them, a numerical method is set up and numerical examples are presented.     

Stability and convergence analysis of a one step approximation of a linear partial integro‐differential equation

We study a linear partial integro‐differential equation which arises in the modeling of various physical and biological sciences. We analyze numerical stability and numerical convergence of a one step approximation of the problem with smooth and non‐smooth initial functions. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1179–1200, 2011     

A new computational approach for solving optimal control of linear PDEs problem

In this paper, we present a new computational approach for solving an internal optimal control problem, which is governed by a linear parabolic partial differential equation. Our approach is to approximate the PDE problem by a nonhomogeneous ordinary differential equation system in higher dimension. Then, the homogeneous part of ODES is solved using semigroup theory. In the next step, the convergence of this approach is verified by means of Toeplitz matrix. In the rest of the paper, the optimal control problem is solved by utilizing the solution of homogeneous part. Finally, a numerical example is given.     

Error Estimates for the Approximation of a Class of Optimal Control Systems Governed by Linear PDEs

This paper deals with a class of optimal control problems in which the system is governed by a linear partial differential equation and the control is distributed and with constraints. The problem is posed in the framework of the theory of optimal control of systems. A numerical method is proposed to approximate the optimal control. In this method, the state space as well as the convex set of admissible controls are discretized. An abstract error estimate for the optimal control problem is obtained that depends on both the approximation of the state equation and the space of controls. This theoretical result is illustrated by some numerical examples from the literature.     

Power Penalty Method for a Linear Complementarity Problem Arising from American Option Valuation

In this paper, we present a power penalty function approach to the linear complementarity problem arising from pricing American options. The problem is first reformulated as a variational inequality problem; the resulting variational inequality problem is then transformed into a nonlinear parabolic partial differential equation (PDE) by adding a power penalty term. It is shown that the solution to the penalized equation converges to that of the variational inequality problem with an arbitrary order. This arbitrary-order convergence rate allows us to achieve the required accuracy of the solution with a small penalty parameter. A numerical scheme for solving the penalized nonlinear PDE is also proposed. Numerical results are given to illustrate the theoretical findings and to show the effectiveness and usefulness of the method. This work was partially supported by a research grant from the University of Western Australia and the Research Grant Council of Hong Kong, Grants PolyU BQ475 and PolyU BQ493.     

Milp modelling for time optimal guidance to a moving target

This paper describes a numerical scheme for optimal control of a time-dependent linear system to a moving final state. Discretization of the corresponding differential equations gives rise to a linear algebraic system. Defining some binary variables, we approximate the original problem by a mixed integer linear programming (MILP) problem. Numerical examples show that the resulting method is highly efficient.     

On the existence and uniqueness of a solution of a linear stochastic differential equation with respect to a logarithmic process

We study the problem of existence and uniqueness of a solution of a linear stochastic differential equation with respect to a logarithmic process. For the conditional mathematical expectation of a solution, we obtain a partial differential equation.