具有无穷阶导数的非线性方程的数值研究
Numerical Study of Nonlinear Equations with Infinite Number of Derivatives
AI总结：
本文通过数值方法研究了一类具有无穷阶导数的非线性方程，发现其解存在插值和周期两种不同模式，并确定了临界参数值q^2=1.37。
摘要：
我们研究了具有无穷多个导数的方程。这类方程构成了数学物理中的一类新方程。这些方程最初出现在p进弦理论中，后来出现在费米子弦理论中，其研究在数学物理及其应用（特别是宇宙学）中具有重要意义。具有无穷多个导数的微分方程可以写成非线性积分方程。我们对这些方程的解进行了数值研究。结果表明，这些方程存在两种不同的解模式：插值模式和周期模式。分离这两种模式的临界参数值被确定为q^2=1.37。我们还证明了这些方程的迭代过程的收敛性。
英文摘要：
We study equations with infinitely many derivatives. Equations of this type form a new class of equations in mathematical physics. These equations originally appeared in p-adic and later in fermionic string theories and their investigation is of much interest in mathematical physics and applications, in particular in cosmology. Differential equation with infinite number of derivatives could be written as nonlinear integral equations. We perform numerical investigation of solutions of the equations. It is established that these equations have two different regimes of the solutions: interpolating and periodic. The critical value of the parameter q separating these regimes is found to be q^2=1.37. Convergence of iterative procedure for these equations is proved.
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