Method of weighted residuals as applied to nonlinear differential equations

A new method for solving a class of nonlinear boundary-value problems is presented. In this method, the nonlinear equation is linearized by guessing an initial solution and using it to evaluate the nonlinear terms. Next, a method of weighted residuals is applied to transform the linearized form of the boundary value problem to an initial value problem. The second (improved) solution is obtained by integrating the initial value problem by a fourth order Runge-Kutta scheme. The entire process is repeated until a desired convergence criterion is achieved.     

Fractional optimal control problem for ordinary differential equation in weighted Lebesgue spaces

Optimization Letters - In this paper, a necessary and sufficient condition, such as the Pontryagin’s maxi-mum principle for a fractional optimal control problem with concentrated parameters,...     

Backward stochastic differential equations and applications to optimal control

We study the existence and uniqueness of the following kind of backward stochastic differential equation, $$x(t) + \int_t^T {f(x(s),y(s),s)ds + \int_t^T {y(s)dW(s) = X,} }$$ under local Lipschitz condition, where (Ω, ?,P, W(·), ?_t) is a standard Wiener process, for any given (x, y),f(x, y, ·) is an ?_t-adapted process, andX is ?_t-measurable. The problem is to look for an adapted pair (x(·),y(·)) that solves the above equation. A generalized matrix Riccati equation of that type is also investigated. A new form of stochastic maximum principle is obtained.     

Efficient error control in numerical integration of ordinary differential equations and optimal interpolating variable-stepsize peer methods

Automatic global error control of numerical schemes is examined. A new approach to this problem is presented. Namely, the problem is reformulated so that the global error is controlled by the numerical method itself rather than by the user. This makes it possible to find numerical solutions satisfying various accuracy requirements in a single run, which so far was considered unrealistic. On the other hand, the asymptotic equality of local and global errors, which is the basic condition of the new method for efficiently controlling the global error, leads to the concept of double quasi-consistency. This requirement cannot be satisfied within the classical families of numerical methods. However, the recently proposed peer methods include schemes with this property. There exist computational procedures based on these methods and polynomial interpolation of fairly high degree that find the numerical solution in a single run. If the integration stepsize is sufficiently small, the error of this solution does not exceed the prescribed tolerance. The theoretical conclusions of this paper are supported by the numerical results obtained for test problems with known solutions.     

Local error control in codes for ordinary differential equations

There are several schemes for the control of local error which are seen in differential equation solvers. The analysis attempts to explain how the selection of a scheme influences the behavior of global error seen in high quality production codes. Two rules of thumb for estimating global errors are given theoretical support when used in conjunction with suitable codes. Substantial numerical experiments support the analysis and conclusions.     

Solution of general ordinary differential equations using the algebraic theory approach

The algebraic theory for numerical methods, as developed by Herrera [3–7], provides a broad theoretical framework for the development and analysis of numerical approximations. To this point, the technique has only been applied to ordinary differential equations with constant coefficients. The present work extends the theory by developing a methodology for equations with variable coefficients. Approximation of the coefficients by piecewise polynomials forms the foundation of the approach. Analysis of the method provides firm error estimates. Furthermore, the analysis points to particular procedures that produce optimal accuracy. Example calculations illustrate the computational procedure and verify the theoretical convergence rates.     

The algebraic theory approach for ordinary differential equations: Highly accurate finite differences

This article reports further developments of Herrera's algebraic theory approach to the numerical treatment of differential equations. A new solution procedure for ordinary differential equations is presented. Finite difference algorithms of 0(h^r), for arbitrary “r” are developed. The method consists in constructing local approximate solutions and using them to extract information about the sought solution. Only nodal information is derived. The local approximate solutions are constructed by collocation, using polynomials of degree G. When “n” collocation points are used at each subinterval, G = n + 1and the order of accuracy is 0(h^2n?1). The procedure here presented is very easy to implement. A program in which n can be chosen arbitrarily, was constructed and applied to selected examples.     

Some third-order ordinary differential equations

The paper deals with periodic orbits in three systems of ordinarydifferential equations. Two of the systems, the Falkner–Skanequations and the Nosé equations, do not possess fixedpoints, and yet interesting dynamics can be found. Here, periodicorbits emerge in bifurcations from heteroclinic cycles, connectingfixed points at infinity. We present existence results for suchperiodic orbits and discuss their properties using careful asymptoticarguments. In the final part results about the Nosé equationsare used to explain the dynamics in a dissipative perturbation,related to a system of dynamo equations.     

The numerical solution of unstable ordinary differential equations

Invariant imbedding, or the Riccati transformation, has been used to solve unstable ordinary differential equations for a few years. This paper compares the above method with parallel or multiple shooting and a method using Chebyshev series. Parallel shooting gives a solution as accurate as that obtained using the Riccati transformation, in a comparable time.     

New approach to optimal control of stochastic Volterra integral equations

ABSTRACT

We study optimal control of stochastic Volterra integral equations (SVIE) with jumps by using Hida-Malliavin calculus.

• We give conditions under which there exist unique solutions of such equations.

• Then we prove both a sufficient maximum principle (a verification theorem) and a necessary maximum principle via Hida-Malliavin calculus.

• As an application we solve a problem of optimal consumption from a cash flow modelled by an SVIE.

     

How to solve the polynomial ordinary differential equations

This paper discussed how to solve the polynomial ordinary differential equations. At first, we construct the theory of the linear equations about the unknown one variable functions with constant coefficients. Secondly, we use this theory to convert the polynomial ordinary differential equations into the simultaneous first order linear ordinary differential equations with constant coefficients and quadratic equations. Thirdly, we work out the general solution of the polynomial ordinary differential equations which is no longer concerned with the differential. Finally, we discuss the necessary and sufficient condition of the existence of the solution.