Related papers: Dealing with Rational Second Order Ordinary Differ…
The purpose of this research is to propose a new approach named the shifted Bessel Tau (SBT) method for solving higher-order ordinary differential equations (ODE). The operational matrices of derivative, integral and product of shifted…
Conventional finite-difference schemes for solving partial differential equations are based on approximating derivatives by finite-differences. In this work, an alternative theory is proposed which view finite-difference schemes as…
The inverse problem of the calculus of variations consists in determining if the solutions of a given system of second order differential equations correspond with the solutions of the Euler-Lagrange equations for some regular Lagrangian.…
We develop a one step matrix method in order to obtain approximate solutions of first order systems and non-linear ordinary differential equations, reducible to first order systems. We find a sequence of such solutions that converge to the…
Uncertainty is unavoidable in modeling dynamical systems and it may be represented mathematically by differential inclusions. In the past, we proposed an algorithm to compute validated solutions of differential inclusions; here we provide…
Properties of partial integrals such as real and complex-valued polynomial, multiple polynomial, exponential, and conditional for ordinary differential systems are studied. The possibilities of constructing first integrals and last…
We develop a well-posedness theory for second order systems in bounded domains where boundary phenomena like glancing and surface waves play an important role. Attempts have previously been made to write a second order system consisting of…
We study, by means of a topological approach, the forced oscillations of second order functional retarded differential equations subject to periodic perturbations. We consider a delay-type functional dependence involving a gamma probability…
We study a complex intertwining relation of second order for Schroedinger operators and construct third order symmetry operators for them. A modification of this approach leads to a higher order shape invariance. We analyze with particular…
We describe an algorithm for the numerical solution of second order linear differential equations in the highly-oscillatory regime. It is founded on the recent observation that the solutions of equations of this type can be accurately…
Results of research of possibility of transformation of a difference equation into a system of the first-order difference equation are presented. In contrast to the method used previously, an unknown grid function is split into two new…
This paper develops methods for numerically solving stochastic delay-differential equations (SDDEs) with multiple fixed delays that do not align with a uniform time mesh. We focus on numerical schemes of strong convergence orders $1/2$ and…
All Darboux integrable difference equations on the quad-graph are described in the case of the equations that possess autonomous first-order integrals in one of the characteristics. A generalization of the discrete Liouville equation is…
This article introduces a framework for measuring the uncertain behaviour of a changing system in terms of the solution of a class of fractional stochastic differential equations (fsDEs). This is accomplished via operational matrices based…
In this paper we propose an algorithm for the numerical solution of arbitrary differential equations of fractional order. The algorithm is obtained by using the following decomposition of the differential equation into a system of…
In this article, firstly we develop a method for a type of difference equations, applicable to solve approximately a class of first order ordinary differential equation systems. In a second step, we apply the results obtained to solve a…
In this paper we construct high order numerical methods for solving third and fourth orders nonlinear functional differential equations (FDE). They are based on the discretization of iterative methods on continuous level with the use of the…
We obtain explicit formulas for the solutions of the system of second-order difference equations of the form $x_{n+ 1} = \frac{x_n y_{n-1}}{y_n (a_n + b_n x_n y_{n - 1})}, \quad y_{n+1} = \frac{x_{n - 1} y_n}{x_n (c_n+d_n x_{n-1} y_n)}$,…
In this paper, we concentrate on solving second-order singularly perturbed Fredholm integro-differential equations (SPFIDEs). It is well known that solving these equations analytically is a challenging endeavor because of the presence of…
We introduce a new numerical method, based on Bernoulli polynomials, for solving multiterm variable-order fractional differential equations. The variable-order fractional derivative was considered in the Caputo sense, while the…