论文标题
一个幽灵扰动方案来解决普通微分方程
A ghost perturbation scheme to solve ordinary differential equations
论文作者
论文摘要
我们提出了一种代数方法，该方法找到了一系列函数，该函数呈指数级接近任何二阶普通微分方程（ODE）的解决方案。我们定义了一个由线性差分运算符组成的扩展ODE（EODE），该差微分操作员依赖于免费参数，$ p $，以及由原始ODE SINUS SINUS组成的$ε$扰动。在EODE的正式$ε$扩展解决方案之后，我们可以通过订单订购线性ODES的订单，并且我们将获得一系列函数$ y_n（x;ε，p）$，其中$ n $表示我们在$ε$ eppection中保留的条款数量。我们通过将$ y_n $的距离函数最小化到给定的$ x $ -Interval上的$ y_n $ $ y_n $，将参数修复到最佳值$ p^*（n）$。我们看到，当$ε= 1 $：$ \ vert y_n（x;ε= 1，p^*（n）） - y（x）\ vert <cδ^^n+1} $ at $Δ<1 $时，EODE的扰动解决方案在$ n $中以$ n $的成倍收敛到ode解决方案。该方法仅通过查看$ n $，$ n $，$ p^{*，α}（n）$的每个顺序的最小函数的最小值来了解边界价值问题的解决方案数量，其中每个$α$都定义了一系列函数$ y_n $，这些函数$ y_n $将转换为ode的解决方案之一。我们通过将其应用于多种情况，在讨论其属性，收益和缺点以及一些实际的算法改进的情况下，将其应用于该方法。
We propose an algebraic method that finds a sequence of functions that exponentially approach the solution of any second-order ordinary differential equation (ODE) with any boundary conditions. We define an extended ODE (eODE) composed of a linear generic differential operator that depends on free parameters, $p$, plus an $ε$ perturbation formed by the original ODE minus the same linear term. After the eODE's formal $ε$ expansion of the solution, we can solve order by order a hierarchy of linear ODEs, and we get a sequence of functions $y_n(x;ε,p)$ where $n$ indicates the number of terms that we keep in the $ε$-expansion. We fix the parameters to the optimal values $p^*(n)$ by minimizing a distance function of $y_n$ to the ODE's solution, $y$, over a given $x$-interval. We see that the eODE's perturbative solution converges exponentially fast in $n$ to the ODE solution when $ε=1$: $\vert y_n(x;ε=1,p^*(n))-y(x)\vert<Cδ^{n+1}$ with $δ<1$. The method permits knowing the number of solutions for Boundary Value Problems just by looking at the number of minima of the distance function at each order in $n$, $p^{*,α}(n)$, where each $α$ defines a sequence of functions $y_n$ that converges to one of the ODE's solutions. We present the method by its application to several cases where we discuss its properties, benefits and shortcomings, and some practical algorithmic improvements.
