Solving a non-local linear differential equation model of the Newtonian-type

IF 1.9 4区 物理与天体物理 Q2 PHYSICS, MULTIDISCIPLINARY Pramana Pub Date : 2024-05-03 DOI:10.1007/s12043-024-02765-8
Wen-Xiu Ma
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Abstract

Motivated by recent studies on non-local integrable models, we consider a non-local inhomogeneous linear differential equation model of Newtonian type:

$$\begin{aligned} \hspace{42pt}x''(t)=\lambda x(t)+\mu x(-t) +f(t),\ t\in {\mathbb {R}}, \end{aligned}$$

where \(\lambda \) and \(\mu \) are real constants and f is continuous. Through decomposing functions into their even and odd parts, we transform the non-local model into a local model, and then with the classical ODE technique, solve the resulting local model under the even and odd constraints. The general solution involving two arbitrary constants is presented in nine cases of the coefficients.

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求解牛顿型非局部线性微分方程模型
受最近关于非局部可积分模型研究的启发,我们考虑了牛顿类型的非局部非均质线性微分方程模型:$$\begin{aligned}.\hspace{42pt}x''(t)=\lambda x(t)+\mu x(-t) +f(t),\t\in {\mathbb {R}}, \end{aligned}$$其中\(\lambda\)和\(\mu\)是实常数,f是连续的。通过将函数分解为偶数部分和奇数部分,我们将非局部模型转化为局部模型,然后利用经典的 ODE 技术,在偶数和奇数约束条件下求解得到的局部模型。在系数的九种情况下,给出了涉及两个任意常数的一般解法。
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来源期刊
Pramana
Pramana 物理-物理:综合
CiteScore
3.60
自引率
7.10%
发文量
206
审稿时长
3 months
期刊介绍: Pramana - Journal of Physics is a monthly research journal in English published by the Indian Academy of Sciences in collaboration with Indian National Science Academy and Indian Physics Association. The journal publishes refereed papers covering current research in Physics, both original contributions - research papers, brief reports or rapid communications - and invited reviews. Pramana also publishes special issues devoted to advances in specific areas of Physics and proceedings of select high quality conferences.
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