Related papers: On the complete integrability and linearization of…
A novel approach is introduced for deriving exact solutions to nonlinear systems of ordinary differential equations. This method consists of four parts. In the initial part, the examined nonlinear differential equation system is transformed…
In this article, using DiPerna-Lions theory \cite{Di-Li}, we investigate linear second order stochastic partial differential equations with unbounded and degenerate non-smooth coefficients, and obtain several conditions for existence and…
A general formalism to solve nonlinear differential equations is given. Solutions are found and reduced to those of second order nonlinear differential equations in one variable. The approach is uniformized in the geometry and solves…
New families of fourth-order composition methods for the numerical integration of initial value problems defined by ordinary differential equations are proposed. They are designed when the problem can be separated into three parts in such a…
Solving inverse and optimization problems over solutions of nonlinear partial differential equations (PDEs) on complex spatial domains is a long-standing challenge. Here we introduce a method that parameterizes the solution using spectral…
In this paper the factorization method introduced by Rosu \& Cornejo-P\'erez for second order non linear differential equations is generalized by adding a parameter in order to obtain the general solutions for the mixed quadratic and linear…
We provide the details of an implementation of Fourier techniques for solving second-order linear partial differential equations (with constant coefficients) using a computer algebra system. The general Sturm-Liouville problem for the heat,…
Solving partial differential equations (PDEs) using an annealing-based approach involves solving generalized eigenvalue problems. Discretizing a PDE yields a system of linear equations (SLE). Solving an SLE can be formulated as a general…
Modal synthesis methods are a long-standing approach for modelling distributed musical systems. In some cases extensions are possible in order to handle geometric nonlinearities. One such case is the high-amplitude vibration of a string,…
We propose a method for transformating linear and nonlinear hypersingular integral equations into ordinary differential equations. Linear and nonlinear polyhypersingular integral equations are transformed into partial differential…
We discuss algorithms applicable to the numerical solution of second-order ordinary differential equations by finite-differences. We make particular reference to the solution of the dissipative particle dynamics fluid model, and present…
In this letter, we introduce a new generalized linearizing transformation (GLT) for second order nonlinear ordinary differential equations (SNODEs). The well known invertible point (IPT) and non-point transformations (NPT) can be derived as…
The explicit integrability of second order ordinary differential equations invariant under time-translation and rescaling is investigated. Quadratic systems generated from the linearisable version of this class of equations are analysed to…
In this paper we construct nonlinear partial differential equations in more than 3 independent variables, possessing a manifold of analytic solutions with high, but not full, dimensionality. For this reason we call them ``partially…
It is well known that second order homogeneous linear ordinary differential equations with slowly varying coefficients admit slowly varying phase functions. This observation underlies the Liouville-Green method and many other techniques for…
The main subject of this paper is the study of analytic second order linear partial differential equations. We aim to solve the classical equations and some more, in the real or complex analytical case. This is done by introducing methods…
Invariant linearization criteria of square systems of second-order quadratically semi-linear ordinary differential equations (ODEs) that can be represented as geodesic equations are extended to square systems of ODEs cubically nonlinear in…
There are many methods for finding a particular solution to a nonhomogeneous linear ordinary differential equation (ODE) with constant coefficients. The method of undetermined coefficients, Laplace transform method and differential operator…
We present a continuous finite element method for some examples of fully nonlinear elliptic equation. A key tool is the discretisation proposed in Lakkis & Pryer (2011, SISC) allowing us to work directly on the strong form of a linear PDE.…
We tersely review a recently introduced technique to identify systems of two nonlinearly-coupled Ordinary Di{\S}erential Equations (ODEs) solvable by algebraic operations; and we report some specifc examples of this kind, namely systems of…