The numerical solution of the matrix Riccati differential equation

A numerically stable and fast computational method is given for the solution of the matrix Ricatti differential equation with finite terminal time.     

Numerical solution of the matrix equation AX + XAT+ B = 0

A recursive algorithm is shown to solve the above equation accurately for large (n leq 146), lightly damped (zeta geq 10^{-3}) systems. About2.5n^{2}storage locations are required, and about2.5n^{3}multiplications are performed per recursion, ten recursions being typical.     

The recursive reduced-order numerical solution of the singularlyperturbed matrix differential Riccati equation

Under stability-observability conditions imposed on a singularly perturbed system, an efficient numerical method for solving the corresponding matrix differential Riccati equation is obtained in terms of the reduced-order problems. The order reduction is achieved via the use of the Chang transformation applied to the Hamiltonian matrix of a singularly perturbed linear-quadratic control problem. An efficient numerical recursive algorithm with a quadratic rate of convergence is developed for solving the algebraic equations comprising the Chang transformation     

The accurate numerical solution of the Schrödinger equation with an explicitly time-dependent Hamiltonian

We show how the highly accurate and efficient Constant Perturbation (CP) technique for steady-state Schrödinger problems can be used in the solution of time-dependent Schrödinger problems with explicitly time-dependent Hamiltonians, following a technique suggested by Ixaru (2010). By introducing a sectorwise spatial discretization using bases of accurately CP-computed eigenfunctions of carefully-chosen stationary problems, we deal with the possible highly oscillatory behavior of the wave function while keeping the dimension of the resulting ODE system low. Also for the time-integration of the ODE system a very effective CP-based approach can be used.     

The numerical solution of the telegraph equation by the alternating group explicit (AGE) method

In this paper, the Alternating Group Explicit (AGE) method is developed from a judicious splitting of the implicit equations derived from the finite difference discretisation of the partial differential equations. The resulting equations can be reformulated in a (2?×?2) explicit form resulting in a new stable and efficient method. Finally, the AGE method is used to investigate the numerical solution of the Telegraph equation in various 2D forms.     

A numerical solution of the stochastic discrete algebraic Riccati equation

This article proposes two algorithms for solving a stochastic discrete algebraic Riccati equation which arises in a stochastic optimal control problem for a discrete-time system. Our algorithms are generalized versions of Hewer’s algorithm. Algorithm I has quadratic convergence, but needs to solve a sequence of extended Lyapunov equations. On the other hand, Algorithm II only needs solutions of standard Lyapunov equations which can be solved easily, but it has a linear convergence. By a numerical example, we shall show that Algorithm I is superior to Algorithm II in cases of large dimensions. This work was presented in part at the 13th International Symposium on Artificial Life and Robotics, Oita, Japan, January 31–February 2, 2008     

A new solution to the generalized Sylvester matrix equation

This note deals with the problem of solving the generalized Sylvester matrix equation AV-EVF=BW, with F being an arbitrary matrix, and provides complete general parametric expressions for the matrices V and W satisfying this equation. The primary feature of this solution is that the matrix F does not need to be in any canonical form, and may be even unknown a priori. The results provide great convenience to the computation and analysis of the solutions to this class of equations, and can perform important functions in many analysis and design problems in control systems theory.     

A numerical study of k(3,2) equation

The Korteweg-de Vries (Kdv) equation has been generalized by Rosenau and Hyman [7] to a class of partial differential equations (PDEs) which has solitary wave solution with compact support. These solitary wave solutions are called compactons

Compactons are solitary waves with the remarkable soliton property, that after colliding with other compactons, they reemerge with the same coherent shape. These particle like waves exhibit elastic collision that are similar to the soliton interaction associated with completely integrable systems. The point where two compactons collide are marked by a creation of low amplitude compacton-anticompacton pair. These equations have only a finite number of local conservation laws

In this paper, an implicit finite difference method and a finite element method have been developed to solve the K(3,2) equation. Accuracy and stability of the methods have been studied. The analytical solution and the conserved quantities are used to assess the accuracy of the suggested methods. The umerical results have shown that this compacton exhibits true soliton behavior.     

On the numerical solution of the Burgers's equation

In this paper, we consider the linear heat equation arisen from the Burgers's equation using the Hopf–Cole transformation. Discretization of this equation with respect to the space variable results in a linear system of ordinary differential equations. The solution of this system involves in computing exp(α A)y for some vector y, where A is a large special tridiagonal matrix and α is a positive real number. We give an explicit expression for computing exp(α A)y. Finally, some numerical experiments are given to show the efficiency of the method.     

A numerical solution of the differential equation u″+2u′/r=u−u3

An algorithm for solving the special boundary value problem for the ordinary differential equation of the second order on interval [0, ∞) is proposed. Results obtained by numerical experiments are also given.     

The numerical solution ofẊ = A_{1}X + XA_{2} + D, X(0) = C

An improved method of solving the general matrix differential equationdot{X} = A_{1}X + XA_{2} + D, X(0) = CforXis considered where A1and A2are stable matrices. The algorithm proposed requires only5n^{2}words of memory and converges in approximately43n^{3} mus where μ is the multiplication time of the digital computer andn = max(n_{1},n_{2})whereA_{1} in R^{n_{1} times n_{1}}, A_{2} in R^{n_{2} times n_{2}}. The algorithm is extremely simple to implement.     

A note on the matrix equation A' LA-L = -K

Matrices WiandN_{i},which result from application of Krylov's algorithm to the matricesAandBrelated byA = (B + I)(B - I)^{-1},are shown to be row equivalent, i.e.,N_{i} = M_{i}W_{1}. This result is applied to solution of the Lyapunov matrix equation for discrete-time systems,A'LA - L = -K.     

A new numerical solution ofẊ = A_{1}X + XA_{2} + D, X(0) = C

Another numerical solution of the general matrix differential equationcirc{X}=A_{1}X+XA_{2}+D, X(0)=Cfor X is considered without any stability condition for A1and A2. Like Davison's method, the proposed algorithm requires only some n2words of memory andn_{3multiplications wheren=max(n_{1},n_{2})andA in R^{n_{1} times n_{1}},A_{2} in R^{n_{2} times n_{2}}. This new approach is well suited to solve large and possibly unstable systems. We take the opportunity to run the differential equation for various D. A very efficient technique follows to design the so-called receding horizon control problem.