For linear partial differential equations, there exist well-developed classical methods of their analytical solution, which, in particular, includes the separation of variables, different integral transforms, Fourier series and their generalizations. At the same time, the study of symmetry properties of such equations is important, first of all, for the development of methods of symmetry analys...

In this paper we study the convergence of the semi-implicit and the implicit Euler methods for the time integration of abstract, quasilinear hyperbolic evolution equations. The analytical framework considered here includes certain quasilinear Maxwell’s and wave equations as special cases. Our analysis shows that the Euler approximations are well-posed and convergent of order one. The techniques...

In this study, the density dependent nonlinear reactiondiffusion equation, which arises in the insect dispersal models, is solved using the combined application of differential quadrature method(DQM) and implicit Euler method. The polynomial based DQM is used to discretize the spatial derivatives of the problem. The resulting time-dependent nonlinear system of ordinary differential equations(OD...

In this article, we report results of an experimental study on six iterative methods to compute the transient probabilities of large Markov models: full matrix exponentiation, forward Euler method, explicit Runge-Kutta methods of order 2 and 4 and Adams-Bashforth multi-steps methods of order 2 and 4. We suggest a simple but efficient implementation of these algorithms. We discuss how to tune th...

Composition methods are methods for the integration of ordinary differential equations arising from differential geometry, or more precisely, Lie algebra theory. We apply them here to the simulation of arrays of Chua’s circuits. In these methods, we split the vector field of the array of Chua’s circuits into its linear part and its nonlinear part. We then solve the elementary differential equat...

Three iteration methods are proposed for the computation of eigenvalues and eigenfunctions in the linear stability of solitary waves. These methods are based on iterating certain time evolution equations associated with the linear stability eigenvalue problem. The first method uses the fourth-order Runge–Kutta method to iterate the pre-conditioned linear stability operator, and it usually conve...