The 'trial function method' ( TFM for short) and a routine way in finding traveling,wave solutions to some nonlinear partial differential equations( PDE for short), wer explained. Two types of evolution equati...The 'trial function method' ( TFM for short) and a routine way in finding traveling,wave solutions to some nonlinear partial differential equations( PDE for short), wer explained. Two types of evolution equations are studied, one is a generalized Burgers or KdV equation, the other is the Fisher equation with special nonlinear forms of its reaction rate term. One can see that this method is simple, fast and allowing further extension.展开更多
In this paper, we propose an accelerated search-extension method （ASEM） based on the interpolated coefficient finite element method, the search-extension method （SEM） and the two-grid method to obtain the multiple...In this paper, we propose an accelerated search-extension method （ASEM） based on the interpolated coefficient finite element method, the search-extension method （SEM） and the two-grid method to obtain the multiple solutions for semilinear elliptic equations. This strategy is not only successfully implemented to obtain multiple solutions for a class of semilinear elliptic boundary value problems, but also reduces the expensive computation greatly. The numerical results in I-D and 2-D cases will show the efficiency of our approach.展开更多
In this paper, we present an effective meshless method for solving the inverse heat conduction problems, with the Neumann boundary condition. A PDE-constrained optimization method is developed to get a global approxim...In this paper, we present an effective meshless method for solving the inverse heat conduction problems, with the Neumann boundary condition. A PDE-constrained optimization method is developed to get a global approximation scheme in both spatial and temporal domains, by using the fundamental solution of the governing equation as the basis function.Since the initial measured data contain some noises, and the resulting systems of equations are usually ill-conditioned, the Tikhonov regularization technique with the generalized crossvalidation criterion is applied to obtain more stable numerical solutions. It is shown that the proposed schemes are effective by some numerical tests.展开更多
This article will combine the finite element method, the interpolated coefficient finite element method, the eigenfunction expansion method, and the search-extension method to obtain the multiple solutions for semilin...This article will combine the finite element method, the interpolated coefficient finite element method, the eigenfunction expansion method, and the search-extension method to obtain the multiple solutions for semilinear elliptic equations. This strategy not only grently reduces the expensive computation, but also is successfully implemented to obtain multiple solutions for a class of semilinear elliptic boundary value problems with non-odd nonlinearity on some convex or nonconvex domains. Numerical solutions illustrated by their graphics for visualization will show the efficiency of the approach.展开更多
In this paper a meshless method of lines is proposed for the numerical solution of time-dependent nonlinear coupled partial differential equations. Contrary to mesh oriented methods of lines using the finite-differenc...In this paper a meshless method of lines is proposed for the numerical solution of time-dependent nonlinear coupled partial differential equations. Contrary to mesh oriented methods of lines using the finite-difference and finite element methods to approximate spatial derivatives, this new technique does not require a mesh in the problem domain, and a set of scattered nodes provided by initial data is required for the solution of the problem using some radial basis functions. Accuracy of the method is assessed in terms of the error norms L2, L∞ and the three invariants C1, C2, C3. Numerical experiments are performed to demonstrate the accuracy and easy implementation of this method for the three classes of time-dependent nonlinear coupled partial differential equations.展开更多
First-order proximal methods that solve linear and bilinear elliptic optimal control problems with a sparsity cost functional are discussed. In particular, fast convergence of these methods is proved. For benchmarking...First-order proximal methods that solve linear and bilinear elliptic optimal control problems with a sparsity cost functional are discussed. In particular, fast convergence of these methods is proved. For benchmarking purposes, inexact proximal schemes are compared to an inexact semismooth Newton method. Results of numerical experiments are presented to demonstrate the computational effectiveness of proximal schemes applied to infinite-dimensional elliptic optimal control problems and to validate the theoretical estimates.展开更多
An object-oriented approach is taken to the problem of formulating portable, easy-to-modify PDE solvers for realistic problems in three space dimensions. The resulting software library, Cogito, contains tools for writ...An object-oriented approach is taken to the problem of formulating portable, easy-to-modify PDE solvers for realistic problems in three space dimensions. The resulting software library, Cogito, contains tools for writing programs to be executed on MIMD computers with distributed memory. Difference methods on composite, structured grids are supported. Most of the Cogito classes have been implemented in Fortran 77, in such a way that the object-oriented design is visible. With respect to parallel performance, these tools yield code that is comparable to parallel solvers written in plain Fortran 77. The resulting programs are can be executed without modification on a large number of multicomputer platforms, and also on serial computers. The uppermost level of abstraction in Cogito concerns the problem of decoupling the numerical method from the PDE problem. The validity of these tools has been preliminarily demonstrated with a C++ implementation for one-dimensional problems.展开更多
A brain tumor occurs when abnormal cells grow, sometimes very rapidly, into an abnormal mass of tissue. The tumor can infect normal tissue, so there is an interaction between healthy and infected cell. The aim of this...A brain tumor occurs when abnormal cells grow, sometimes very rapidly, into an abnormal mass of tissue. The tumor can infect normal tissue, so there is an interaction between healthy and infected cell. The aim of this paper is to propose some efficient and accurate numerical methods for the computational solution of one-dimensional continuous basic models for the growth and control of brain tumors. After computing the analytical solution, we construct approximations of the solution to the problem using a standard second order finite difference method for space discretization and the Crank-Nicolson method for time discretization. Then, we investigate the convergence behavior of Conjugate gradient and generalized minimum residual as Krylov subspace methods to solve the tridiagonal toeplitz matrix system derived.展开更多
Optimal boundary control of semilinear parabolic equations requires efficient solution methods in applications. Solution methods bypass the nonlinearity in different approaches. One approach can be quasilinearization ...Optimal boundary control of semilinear parabolic equations requires efficient solution methods in applications. Solution methods bypass the nonlinearity in different approaches. One approach can be quasilinearization (QL) but its applicability is locally in time. Nonetheless, consecutive applications of it can form a new method which is applicable globally in time. Dividing the control problem equivalently into many finite consecutive control subproblems they can be solved consecutively by a QL method. The proposed QL method for each subproblem constructs an infinite sequence of linear-quadratic optimal boundary control problems. These problems have solutions which converge to any optimal solutions of the subproblem. This implies the uniqueness of optimal solution to the subproblem. Merging solutions to the subproblems the solution of original control problem is obtained and its uniqueness is concluded. This uniqueness result is new. The proposed consecutive quasilinearization method is numerically stable with convergence order at least linear. Its consecutive feature prevents large scale computations and increases machine applicability. Its applicability for globalization of locally convergent methods makes it attractive for designing fast hybrid solution methods with global convergence.展开更多
文摘The 'trial function method' ( TFM for short) and a routine way in finding traveling,wave solutions to some nonlinear partial differential equations( PDE for short), wer explained. Two types of evolution equations are studied, one is a generalized Burgers or KdV equation, the other is the Fisher equation with special nonlinear forms of its reaction rate term. One can see that this method is simple, fast and allowing further extension.
基金supported by the National Natural Science Foundation of China (10571053, 10871066, 10811120282)Programme for New Century Excellent Talents in University(NCET-06-0712)
文摘In this paper, we propose an accelerated search-extension method （ASEM） based on the interpolated coefficient finite element method, the search-extension method （SEM） and the two-grid method to obtain the multiple solutions for semilinear elliptic equations. This strategy is not only successfully implemented to obtain multiple solutions for a class of semilinear elliptic boundary value problems, but also reduces the expensive computation greatly. The numerical results in I-D and 2-D cases will show the efficiency of our approach.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1129014311471066+3 种基金11572081)the Fundamental Research of Civil Aircraft(Grant No.MJ-F-2012-04)the Fundamental Research Funds for the Central Universities(Grant No.DUT15LK44)the Scientific Research Funds of Inner Mongolia University for the Nationalities(Grant No.NMD1304)
文摘In this paper, we present an effective meshless method for solving the inverse heat conduction problems, with the Neumann boundary condition. A PDE-constrained optimization method is developed to get a global approximation scheme in both spatial and temporal domains, by using the fundamental solution of the governing equation as the basis function.Since the initial measured data contain some noises, and the resulting systems of equations are usually ill-conditioned, the Tikhonov regularization technique with the generalized crossvalidation criterion is applied to obtain more stable numerical solutions. It is shown that the proposed schemes are effective by some numerical tests.
基金This research was supported by the National Natural Science Foundation of China (10571053)Scientific Research Fund of Hunan Provincial Education Department (0513039)the Special Funds of State Major Basic Research Projects (G1999032804)
文摘This article will combine the finite element method, the interpolated coefficient finite element method, the eigenfunction expansion method, and the search-extension method to obtain the multiple solutions for semilinear elliptic equations. This strategy not only grently reduces the expensive computation, but also is successfully implemented to obtain multiple solutions for a class of semilinear elliptic boundary value problems with non-odd nonlinearity on some convex or nonconvex domains. Numerical solutions illustrated by their graphics for visualization will show the efficiency of the approach.
文摘In this paper a meshless method of lines is proposed for the numerical solution of time-dependent nonlinear coupled partial differential equations. Contrary to mesh oriented methods of lines using the finite-difference and finite element methods to approximate spatial derivatives, this new technique does not require a mesh in the problem domain, and a set of scattered nodes provided by initial data is required for the solution of the problem using some radial basis functions. Accuracy of the method is assessed in terms of the error norms L2, L∞ and the three invariants C1, C2, C3. Numerical experiments are performed to demonstrate the accuracy and easy implementation of this method for the three classes of time-dependent nonlinear coupled partial differential equations.
文摘First-order proximal methods that solve linear and bilinear elliptic optimal control problems with a sparsity cost functional are discussed. In particular, fast convergence of these methods is proved. For benchmarking purposes, inexact proximal schemes are compared to an inexact semismooth Newton method. Results of numerical experiments are presented to demonstrate the computational effectiveness of proximal schemes applied to infinite-dimensional elliptic optimal control problems and to validate the theoretical estimates.
文摘An object-oriented approach is taken to the problem of formulating portable, easy-to-modify PDE solvers for realistic problems in three space dimensions. The resulting software library, Cogito, contains tools for writing programs to be executed on MIMD computers with distributed memory. Difference methods on composite, structured grids are supported. Most of the Cogito classes have been implemented in Fortran 77, in such a way that the object-oriented design is visible. With respect to parallel performance, these tools yield code that is comparable to parallel solvers written in plain Fortran 77. The resulting programs are can be executed without modification on a large number of multicomputer platforms, and also on serial computers. The uppermost level of abstraction in Cogito concerns the problem of decoupling the numerical method from the PDE problem. The validity of these tools has been preliminarily demonstrated with a C++ implementation for one-dimensional problems.
文摘A brain tumor occurs when abnormal cells grow, sometimes very rapidly, into an abnormal mass of tissue. The tumor can infect normal tissue, so there is an interaction between healthy and infected cell. The aim of this paper is to propose some efficient and accurate numerical methods for the computational solution of one-dimensional continuous basic models for the growth and control of brain tumors. After computing the analytical solution, we construct approximations of the solution to the problem using a standard second order finite difference method for space discretization and the Crank-Nicolson method for time discretization. Then, we investigate the convergence behavior of Conjugate gradient and generalized minimum residual as Krylov subspace methods to solve the tridiagonal toeplitz matrix system derived.
文摘Optimal boundary control of semilinear parabolic equations requires efficient solution methods in applications. Solution methods bypass the nonlinearity in different approaches. One approach can be quasilinearization (QL) but its applicability is locally in time. Nonetheless, consecutive applications of it can form a new method which is applicable globally in time. Dividing the control problem equivalently into many finite consecutive control subproblems they can be solved consecutively by a QL method. The proposed QL method for each subproblem constructs an infinite sequence of linear-quadratic optimal boundary control problems. These problems have solutions which converge to any optimal solutions of the subproblem. This implies the uniqueness of optimal solution to the subproblem. Merging solutions to the subproblems the solution of original control problem is obtained and its uniqueness is concluded. This uniqueness result is new. The proposed consecutive quasilinearization method is numerically stable with convergence order at least linear. Its consecutive feature prevents large scale computations and increases machine applicability. Its applicability for globalization of locally convergent methods makes it attractive for designing fast hybrid solution methods with global convergence.
