Related papers: On Solving Some Classes of Second Order ODEs
A subroutine for very-high-precision numerical solution of a class of ordinary differential equations is provided. For given evaluation point and equation parameters the memory requirement scales linearly with precision $P$, and the number…
This article introduces a pedagogical method for {\it solving combinatorial problems} that frequently involve structures that are unfamiliar or less familiar. Indeed, an indirect method has been proposed in order to evade any possible…
Developing algorithms for solving high-dimensional partial differential equations (PDEs) has been an exceedingly difficult task for a long time, due to the notoriously difficult problem known as the "curse of dimensionality". This paper…
This work presents a brief discussion and a plan towards the analytical solving of Partial Differential Equations (PDEs) using symbolic computing, as well as an implementation of part of this plan as the PDEtools software-package of…
This note reports on the recent advancements in the search for explicit representation, in classical special functions, of the solutions of the fourth-order ordinary differential equations named Bessel-type, Jacobi-type, Laguerre-type,…
In this paper, we have considered second order non-homogeneous linear differential equations having entire coefficients. We have established conditions ensuring non-existence of finite order solution of such type of differential equations.
The application of the approximation-operational approach to solving linear differential equations of fractional order with variable coefficients is considered. It is shown that the method can also be applied to solving differential…
The quest for analytical solutions to differential equations has traditionally been constrained by the need for extensive mathematical expertise. Machine learning methods like genetic algorithms have shown promise in this domain, but are…
We consider existence of periodic boundary value problems of nonlinear second order ordinary differential equations. Under certain half Lipschitzian type conditions several existence results are obtained. As applications positive periodic…
We propose a physical analogy between finding the solution of an ordinary differential equation (ODE) and a $N$ particle problem in statistical mechanics. It uses the fact that the solution of an ODE is equivalent to obtain the minimum of a…
We consider semiclassical orthogonal polynomials on the unit circle associated with a weight function that satisfy a Pearson-type differential equation involving two polynomials of degree at most three. Structure relations and difference…
A new method of solving third-order ordinary complex differential equations (OCDEs) by generalizing Prelle-Singer. The idea which is a procedure for finding the solution for second-order differential equations in the real domain. We have…
The goal of this work is to characterize all second order difference operators of several variables that have discrete orthogonal polynomials as eigenfunctions. Under some mild assumptions, we give a complete solution of the problem.
In this article we study a system of eikonal equations. Our aim is to isolate the solutions which minimise the discontinuity set of the gradient.
In in this paper we show how using D.A. it is found a simple change of variables (c.v.) that brings us to obtain differential equations simpler than the original one. In a pedagogical way (at least we try to do that) and in order to make…
The aim of the present paper is to propose an algorithm for a new ODE--solver which should improve the abilities of current solvers to handle second order differential equations. The paper provides also a theoretical result revealing the…
We design and analyse a new numerical method to solve ODE system based on the structural method. We compute approximations of solutions together with its derivatives up to order $K$ by solving an entire block corresponding to $R$ time…
Second order ordinary differential equations of the form $y'' = P(x,y) + 4 Q(x,y) y' + 6 R(x,y) y'^2 + 4 S(x,y) y'^3 + L(x,y) y'^4$ are considered and their point-expansions are constructed. Geometrical structures connected with these…
The Lagrangian formalism for variational problem for second-order delay ordinary differential equations (DODEs) is developed. The Noether-type operator identities and theorems for DODEs of second order are presented. Algebraic construction…
In this paper, numerical methods using Physics-Informed Neural Networks (PINNs) are presented with the aim to solve higher-order ordinary differential equations (ODEs). Indeed, this deep-learning technique is successfully applied for…