K. A. Nguyen, M. van der Put
The theme of this paper is to `solve' an absolutely irreducible differential module explicitly in terms of modules of lower dimension and finite extensions of the differential field $K$. Representations of semi-simple Lie algebras and differential Galois theory are the main tools. The results extend the classical work of G. Fano.
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In this paper we presents an algorithm for finding a solution of the linear nonhomogeneous quaternionic-valued differential equations. Moveover, several examples shows the feasibility of our algorithm.
In the present paper authors introduce the L_n-integral transform and the inverse integral transform for n = 2^k, k=0,1,2,..., as a generalization of the classical Laplace transform and the inverse Laplace transform, respectively. Applicability of this transforms in solving linear ordinary differential equations is analyzed. Some illustrative examples are also given.
We present a new approach to solving polynomial ordinary differential equations by transforming them to linear functional equations and then solving the linear functional equations. We will focus most of our attention upon the first-order Abel differential equation with two nonlinear terms in order to demonstrate in as much detail as possible the computations necessary for a complete solution. We mention in our section on further developments that the basic transformation idea can be generalized to apply to differential equations of any order, to a system of ordinary differential equations wit…
An overview of the solution methods for ordinary differential equations in the Mathematica function DSolve is presented.
In this work we consider a given root of a family of n-degree polynomials as a one-variable function that depends only on the independent term. Then we prove that this function satisfies several ordinary differential equations (ODE). More concretely, it satisfies several simple separated variables ODE, a first order generalized Abel ODE of degree n-1 and an (n-1)-th order linear ODE. Although some of our results are not new, our approach is simple and self-contained. For n=2, 3 and 4 we recover, from these ODE, the classical formulas for solving these polynomials.
$D$-Module Techniques for Solving Differential Equations in the Context of Feynman Integrals
本文以三角形积分为例，比较$D$-模方法与专用方法求解费曼积分相关微分方程，实现Saito-Sturmfels-Takayama算法推导标准级数解并与Fuchs系统渐近级数对比。
In this article, we present an innovative method to find particular solutions of the non-homogeneous Cauchy-Euler equations in several variables and Sturm-Liouville equations. The method is basically built on the application of the ponderation ring which introduced by Assal and Zeyada [2].
In this paper, we introduce some analytical techniques to solve some classes of second order differential equations. Such classes of differential equations arise in describing some mathematical problems in Physics and Engineering.
Solving conformable Gegenbauer differential equation and exploring its generating function
本文研究 conformable Gegenbauer 微分方程的求解，验证其解与幂级数结果一致，并证明相关函数性质、递归关系、conformable Rodriguez 公式及生成函数。
The real numbers, it is taught at universities, correspond to our idea of a continuum, although the hyperreal numbers are located ``in between'' the real numbers. The number $x + dx$, where $dx$ should be an infinitesimal number and $x$ real, is infinitesimally close to $x$ but ``infinitely'' far away from all other real numbers. Analogously: If $f'(x_0)$ and $f(x_0)$ are given for a differentiable function $f:\mathbb{R}\rightarrow\mathbb{R}$ at $x_0\in\mathbb{R}$, we can not determine $f(x)$ at {\em any} point $x\in \mathbb{R}$ different from $x_0$. These points seem to be ``infinitely'' far …
Review of implicit methods of integrating system of stiff ordinary differential equations is presented. Defines and graphically presents absolute stability region for Gears methods (backward differentiation formula) used to solve system of stiff ordinary differential equations. Recommendations for selecting the order of Gears method are given.
An effective method to obtain exact analytical solutions of equations describing the coherent dynamics of multilevel systems is presented. The method is based on the usage of orthogonal polynomials, integral transforms and their discrete analogues. All the obtained solutions are expressed by way of special or elementary functions.
Solving first order differential equations presenting elementary functions
本研究通过建立含初等函数的一阶微分方程与带刘维尔首次积分的有理二阶常微分方程的关联，拓展S函数方法相关成果，提出可高效处理混沌系统、确定可积区域的一阶微分方程积分方法。
A technique which is known as Sumudu Transform Method (STM) is studied for the construction of solutions of a most general form of delay differential equations of pantograph type. This is a pioneer study on using the STM to construct the solutions of delay differential equations of pantograph type with variable coefficients. We obtain the exact and approximate solutions of nonlinear problems with multiproportional delays and variable coefficients. The strength of STM is illustrated in reducing the complex computational work as compared to the well-known methods. This paper shows how to succinc…
We provide in this work a robust solution theory for random rough differential equations of mean field type $$ dX_t = V(X_t,\mathcal{L}(X_t))dt + F(X_t,\mathcal{L}(X_t))dW_t, $$ where $W$ is a random rough path and $\mathcal{L}(X_t)$ stands for the law of $X_t$, with mean field interaction in both the drift and diffusivity. The analysis requires the introduction of a new rough path-like setting and an associated notion of controlled path. We use crucially Lions' approach to differential calculus on Wasserstein space along the way.
A Numerical Approach for Solving of Fractional Emden-Fowler Type Equations
本文提出利用分数阶微分变换（FDT）求解分数阶 Emden-Fowler 型微分方程的奇异初值问题，将解表示为快速可计算的收敛级数，数值实验验证了该方法正确、准确且易于实现。
A new semi-analytical approach for numerical solving of Cauchy problem for functional differential equations
本文提出了一种新的半解析方法，结合了分步法和微分变换方法（DTM），用于求解具有延迟或中性微分方程的Cauchy问题，该方法无需符号计算或初始猜测，且在多个常数延迟的FDE类上表现出高效性。
Novel hybrid function operational matrices of fractional integration: An application for solving multi-order fractional differential equations
本文提出一种基于新型正交混合函数的数值算法，用于求解多阶分数微分方程，通过比较实验验证了算法的准确性和效率。
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