Solving a class of nonautonomous difference equations by generalized invariants

By using generalized invariants, we describe a method for solving a class of higher‐order nonautonomous difference equations. Solvability of the equations of order two, three, and four are studied in detail. Our results explain and extend some problems in the literature. As far as we know, the case when the order is four is considered for the first time in this paper. For the equations of second order, we also give an explanation how they can be obtained in a natural way.     

Global well-posedness for a drug transport model in tumor multicell spheroids

In this paper we study global existence of solutions of a mathematical model for drug transport in tumor multicell spheroids. The model is a free boundary problem of a system of partial differential equations. It contains one nonlinear first-order equation describing the distribution of live tumor cells, and two nonlinear reaction diffusion equations describing the evolution of nutrient concentration and drug concentration, respectively. By using the method of characteristics for first-order equations, the $L^p$-theory for parabolic equations, the Banach fixed point theorem and the extension method, we prove that this problem has a unique global solution.     

A matrix rational Lanczos method for model reduction in large‐scale first‐ and second‐order dynamical systems

In the present paper, we describe an adaptive modified rational global Lanczos algorithm for model‐order reduction problems using multipoint moment matching‐based methods. The major problem of these methods is the selection of some interpolation points. We first propose a modified rational global Lanczos process and then we derive Lanczos‐like equations for the global case. Next, we propose adaptive techniques for choosing the interpolation points. Second‐order dynamical systems are also considered in this paper, and the adaptive modified rational global Lanczos algorithm is applied to an equivalent state space model. Finally, some numerical examples will be given.     

Oscillation and non‐oscillation results for solutions of perturbed half‐linear equations

The purpose of this paper is to describe the oscillatory properties of second‐order Euler‐type half‐linear differential equations with perturbations in both terms. All but one perturbations in each term are considered to be given by finite sums of periodic continuous functions, while coefficients in the last perturbations are considered to be general continuous functions. Since the periodic behavior of the coefficients enables us to solve the oscillation and non‐oscillation of the considered equations, including the so‐called critical case, we determine the oscillatory properties of the equations with the last general perturbations. As the main result, we prove that the studied equations are conditionally oscillatory in the considered very general setting. The novelty of our results is illustrated by many examples, and we give concrete new corollaries as well. Note that the obtained results are new even in the case of linear equations.     

Distributed optimal control of the viscous Burgers equation via a Legendre pseudo‐spectral approach

This paper presents a computational technique based on the pseudo‐spectral method for the solution of distributed optimal control problem for the viscous Burgers equation. By using pseudo‐spectral method, the problem is converted to a classical optimal control problem governed by a system of ordinary differential equations, which can be solved by well‐developed direct or indirect methods. For solving the resulting optimal control problem, we present an indirect method by deriving and numerically solving the first‐order optimality conditions. Numerical tests involving both unconstrained and constrained control problems are considered. Copyright © 2015 John Wiley & Sons, Ltd.     

A numerical study of parabolic problems with nonlinear boundary conditions

In this paper, we consider an initial‐boundary value problem for a parabolic equation with nonlinear boundary conditions. The solution to the problem can be expressed as a convolution integral of a Green's function and two unknown functions. We change the problem to a system of two nonlinear Volterra integral equations of convolution type. By using an explicit procedure on the basis of Sinc‐function properties, the resulting integral equations are replaced by a system of nonlinear algebraic equations, whose solution yields an accurate approximate solution to the parabolic problem. Some examples are considered to illustrate the ability of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.     

Optimal Control Problem for Cancer Invasion Reaction–Diffusion System

Abstract

An optimal control problem constrained by a reaction–diffusion mathematical model which incorporates the cancer invasion and its treatment is considered. The state equations consisting of three unknown variables namely tumor cell density, normal cell density, and drug concentration. The main goal of the considered optimal control problem is to minimize the density of cancer cells and decreasing the side effects of treatment. Moreover, existence of a weak solution of brain tumor reaction–diffusion system and the corresponding adjoint system of optimal control problem is also investigated. Further, existence of minimizer for the optimal control problem is established and also the first-order optimality conditions are derived.     

A modified Chebyshev ϑ‐weighted Crank–Nicolson method for analyzing fractional sub‐diffusion equations

This study presents a robust modification of Chebyshev ? ‐weighted Crank–Nicolson method for analyzing the sub‐diffusion equations in the Caputo fractional sense. In order to solve the problem, by discretization of the sub‐fractional diffusion equations using Taylor's expansion a linear system of algebraic equations that can be analyzed by numerical methods is presented. Furthermore, consistency, convergence, and stability analysis of the suggested method are discussed. In this framework, compact structures of sub‐diffusion equations are considered as prototype examples. The main advantage of the proposed method is that, it is more efficient in terms of CPU time, computational cost and accuracy in comparing with the existing ones in open literature.     

Mathematical model of optimal chemotherapy strategy with allowance for cell population dynamics in a heterogeneous tumor

A mathematical model of tumor cell population dynamics is considered. The tumor is assumed to consist of cells of two types: amenable and resistant to chemotherapeutic treatment. It is assumed that the growth of the cell populations of both types is governed by logistic equations. The effect of a chemotherapeutic drug on the tumor is specified by a therapy function. Two types of therapy functions are considered: a monotonically increasing function and a nonmonotone one with a threshold. In the former case, the effect of a drug on the tumor is stronger at a higher drug concentration. In the latter case, a threshold drug concentration exists above which the effect of the therapy reduces. The case when the total drug amount is subject to an integral constraint is also studied. A similar problem was previously studied in the case of a linear therapy function with no constraint imposed on the drug amount. By applying the Pontryagin maximum principle, necessary optimality conditions are found, which are used to draw important conclusions about the character of the optimal therapy strategy. The optimal control problem of minimizing the total number of tumor cells is solved numerically in the case of a monotone or threshold therapy function with allowance for the integral constraint on the drug amount.     

Multipole fast algorithm for the least‐squares approach of the method of fundamental solutions for three‐dimensional harmonic problems

In this article we describe an improvement in the speed of computation for the least‐squares method of fundamental solutions (MFS) by means of Greengard and Rokhlin's FMA. Iterative solution of the linear system of equations is performed for the equations given by the least‐squares formulation of the MFS. The results of applying the method to test problems from potential theory with a number of boundary points in the order of 80,000 show that the method can achieve fast solutions for the potential and its directional derivatives. The results show little loss of accuracy and a major reduction in the memory requirements compared to the direct solution method of the least squares problem with storage of the full MFS matrix. The method can be extended to the solution of overdetermined systems of equations arising from boundary integral methods with a large number of boundary integration points. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 828–845, 2003.     

Stroboscopic averaging of highly oscillatory nonlinear wave equations

In this paper, we are concerned with stroboscopic averaging for highly oscillatory evolution equations posed in a Banach space. Using Taylor expansion, we construct a non‐oscillatory high‐order system whose solution remains exponentially close to the exact one over a long time. We then apply this result to the nonlinear wave equation in one dimension. We present the stroboscopic averaging method, which is a numerical method introduced by Chartier, Murua and Sanz‐Serna, and apply it to our problem. Finally, we conclude by presenting the qualitative and quantitative efficiency of this numerical method for some nonlinear wave problem. Copyright © 2014 John Wiley & Sons, Ltd.     

Constructing rational and multi‐wave solutions to higher order NEEs via the Exp‐function method

In this paper, we present an application of some known generalizations of the Exp‐function method to the fifth‐order Burgers and to the seventh‐order Korteweg de Vries equations for the first time. The two examples show that the Exp‐function method can be an effective alternative tool for explicitly constructing rational and multi‐wave solutions with arbitrary parameters to higher order nonlinear evolution equations. Being straightforward and concise, as pointed out previously, this procedure does not require the bilinear representation of the equation considered. Copyright © 2011 John Wiley & Sons, Ltd.     

Error estimations for some meshless boundary interpolation methods

Meshless methods have become quite popular in numerical treatment of partial differential equations because of their simplicity and the fact that they require neither domain nor boundary mesh. In general, however, they convert the original problem to a highly ill‐conditioned linear system of algebraic equations with a dense matrix. Recently, a special technique has been proposed which circumvents this computational difficulty. This method, called Direct Multi‐Elliptic Interpolation Method, is based on a scattered data interpolation which defines the interpolation function as a solution of a higher order multi‐elliptic equation. Here the boundary version of this meshless method which is based on a multi‐elliptic boundary interpolation is considered. Error estimations are derived justifying the interpolation function to be a good approximation of the solution of the original boundary value problem as well. At the same time, the problem of large, dense and ill‐conditioned matrices as well as the mesh generation are completely avoided. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Exterior electromagnetic shaping using wavelet BEM

The present paper is devoted to exterior electromagnetic shaping in two dimensions. We model the conductors by regular densities which leads to a finite objective and allows a line‐search. In order to compute the surface pressure we optimize an Augmented Lagrangian by a Newton method using a second‐order approach for the Lagrange multiplier. Since the underlying state function satisfies an exterior boundary value problem, we compute first and second order derivatives of its boundary data by boundary integral equations which are numerically solved by a fast wavelet Galerkin scheme. Numerical results prove that we succeeded in finding a fast and robust algorithm for solving the considered class of problems. Copyright © 2004 John Wiley & Sons, Ltd.     

Optimal control synthesis in therapy of solid tumor growth

A mathematical model of tumor growth therapy is considered. The total amount of a drug is bounded and fixed. The problem is to choose an optimal therapeutic strategy, i.e., to choose an amount of the drug permanently affecting the tumor that minimizes the number of tumor cells by a given time. The problem is solved by the dynamic programming method. Exact and approximate solutions to the corresponding Hamilton-Jacobi-Bellman equation are found. An error estimate is proved. Numerical results are presented.     

Numerical solution of fractional differential equations with a Tau method based on Legendre and Bernstein polynomials

In this paper, we state and prove a new formula expressing explicitly the integratives of Bernstein polynomials (or B‐polynomials) of any degree and for any fractional‐order in terms of B‐polynomials themselves. We derive the transformation matrices that map the Bernstein and Legendre forms of a degree‐n polynomial on [0,1] into each other. By using their transformation matrices, we derive the operational matrices of integration and product of the Bernstein polynomials. These matrices together with the Tau method are then utilized to reduce the solution of this problem to the solution of a system of algebraic equations. The method is applied to solve linear and nonlinear fractional differential equations. Copyright © 2013 John Wiley & Sons, Ltd.     

New exact solutions for seventh‐order Sawada–Kotera–Ito,Lax and Kaup–Kupershmidt equations using Exp‐function method

In this work, Exp‐function method is used to solve three different seventh‐order nonlinear partial differential KdV equations. Sawada–Kotera–Ito, Lax and Kaup–Kupershmidt equations are well known and considered for solve. Exp‐function method can be used as an alternative to obtain analytic and approximate solutions of different types of differential equations applied in engineering mathematics. Ultimately this method is implemented to solve these equations and convenient and effective solutions are obtained. Copyright © 2009 John Wiley & Sons, Ltd.     

A discontinuous interpolated finite volume approximation of semilinear elliptic optimal control problems

In this article, we describe a discontinuous finite volume method with interpolated coefficients for the numerical approximation of the distributed optimal control problem governed by a class of semilinear elliptic equations with control constraints. The proposed distributed control problem involves three unknown variable: control, state and costate. For the approximation of control, we have adopted three different methodologies: variational discretization, piecewise constant and piecewise linear discretization, while the approximation of state and costate variables is based on discontinuous piecewise linear polynomials. As the resulted scheme is non‐symmetric, optimize‐then‐discretize approach is used to approximate the control problem. Optimal a priori error estimates in suitable natural norms for state, costate and control variables are derived. Moreover, numerical experiments are presented to support the derived theoretical results. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2090–2113, 2017     

A block preconditioning technique for the streamfunction‐vorticity formulation of the Navier‐Stokes equations

Iterative methods of Krylov‐subspace type can be very effective solvers for matrix systems resulting from partial differential equations if appropriate preconditioning is employed. We describe and test block preconditioners based on a Schur complement approximation which uses a multigrid method for finite element approximations of the linearized incompressible Navier‐Stokes equations in streamfunction and vorticity formulation. By using a Picard iteration, we use this technology to solve fully nonlinear Navier‐Stokes problems. The solvers which result scale very well with problem parameters. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Local interpolants for Numerov‐type methods and their implementation in variable step schemes

The classical explicit fourth‐order Numerov‐type method is considered. The equations of condition for deriving the corresponding interpolants are given. Then using a local error estimation, we may construct a stable variable step scheme. Applying this new implementation in a set of problems, we get very pleasant results.