Solving nonlinear oscillations is a challenging task due to the mathematical complexity of related differential equations. In many cases, determining oscillation’s period requires solution complicated integrals using numerical methods. To avoid complexity, there are empirical equations in literature that can be used instead rigorous analysis provide an acceptable approximation. this paper, rece...

We investigate the structure of Polchinski’s formulation of the flow equations for the continuum Wilson effective action. Reinterpretations in terms of I.R. cutoff greens functions are given. A promising non-perturbative approximation scheme is derived by carefully taking the sharp cutoff limit and expanding in ‘irrelevancy’ of operators. We illustrate with two simple models of four dimensional...

Linear Methods are often used to compute approximate solutions to dynamic models, as these models often cannot be solved analytically. Linear methods are very popular, as they can easily be implemented. Also, they provide a useful starting point for understanding more elaborate numerical methods. It shall be described here first for the example of a simple real business cycle model, including h...

Problems of optimal structural design for minimum deflection, maximum fundamental frequency and maximum stability are considered in this note. The optimality conditions for these problems consist of nonlinear differential equations which usually require a numerical solution [1-16]. An alternate approach based on the Ritz method is presented here, in which one works with an approximate expressio...

In the present paper, we propose numerical methods for solving the Stein equation AXC − X − D = 0 where the matrix A is large and sparse. Such problems appear in discrete-time control problems, filtering and image restoration. We consider the case where the matrix D is of full rank and the case where D is factored as a product of two matrices. The proposed methods are Krylov subspace methods ba...