Related papers: On singular Frobenius for linear differential equa…
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…
Non-local equations cannot be treated using classical ODE theorems. Nevertheless, several new methods have been introduced in the non-local gluing scheme of our previous article "On higher dimensional singularities for the fractional Yamabe…
We consider the phase-integral method applied to an arbitrary linear ordinary second-order differential equation with non-analytical coefficients. We propose a universal technique based on the Frobenius method which allows to obtain new…
In this article, we study about the solutions of second order linear differential equations by considering several conditions on the coefficients of homogenous linear differential equation and its associated non-homogenous linear…
We solve the local equivalence problem for second order (smooth or analytic) ordinary differential equations. We do so by presenting a {\em complete convergent normal form} for this class of ODEs. The normal form is optimal in the sense…
We consider fractional differential equations of order $\alpha \in (0,1)$ for functions of one independent variable $t\in (0,\infty)$ with the Riemann-Liouville and Caputo-Dzhrbashyan fractional derivatives. A precise estimate for the order…
We solve some forms of non homogeneous differential equations in one and two dimensions. By expanding the solution into whell-posed closed form-Eisenstein series the solution itself is quite simple and elementary. Also we consider Fourier…
We study the local classification of higher order Fuchsian linear differential equations under various refinements of the classical notion of the "type of differential equation" introduced by Frobenius. The main source of difficulties is…
The objective of this paper is to complete certain issues from our recent contribution [J. Calatayud, J.-C. Cort\'es, M. Jornet, L. Villafuerte, Random non-autonomous second order linear differential equations: mean square analytic…
Identifying integrable coupled nonlinear ordinary differential equations (ODEs) of dissipative type and deducing their general solutions are some of the challenging tasks in nonlinear dynamics. In this paper we undertake these problems and…
Transformations of differential equations to other equivalent equations play a central role in many routines for solving intricate equations. A class of differential equations that are particularly amenable to solution techniques based on…
These notes aim to provide a classical approach to solving some conformable differential equations based on prior knowledge of how to solve ordinary differential equations. That is, using the methods of separation of variables, homogeneous…
We investigate and derive second solutions to linear homogeneous second-order difference equations using a variety of methods, in each case going beyond the purely formal solution and giving explicit expressions for the second solution. We…
We have already dealt with the problem of solving First Order Differential Equations (1ODEs) presenting elementary functions before in [1, 2]. In this present paper, we have established solid theoretical basis through a relation between the…
In this paper, we present new techniques for solving a large variety of partial differential equations. The proposed method reduces the PDEs to first order differential equations known as classical equations such as Bernoulli, Ricatti and…
We show that for every second order Fuchsian linear differential equation $E$ with $n$ singularities of which $n-3$ are apparent there exists a hypergeometric equation $H$ and a linear differential operator with polynomial coefficients…
A new analytical operator method is discussed which solves linear ordinary differential equations with regular singularities. Solutions are obtained in analytic series form and also in Mellin-Barnes-type contour integral form. Exact series…
In this paper, using the similarity method, we construct particular solutions with singularities for degenerate high-order equations. The considered equations have singularities of the first and second kind. Particular solutions are…
In this note, we establish a new closed formula for the solution of homogeneous second-order linear difference equations with constant coefficients by using matrix theory. This, in turn, gives new closed formulas concerning all sequences of…
The study of existence and uniqueness of solutions became important due to the lack of general formula for solving nonlinear ordinary differential equations (ODEs). Compact form of existence and uniqueness theory appeared nearly 200 years…