Solving differential equations by a maximum entropy-minimum norm method with applications to Fokker-Planck equations
Solving differential equations by a maximum entropy-minimum norm method with applications to Fokker-Planck equations
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DOI：
10.1063/1.528276
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发表时间：
1989-07
影响因子：
1.3
通讯作者：
J. Baker-Jarvis;M. Racine;Jihad Alameddine
中科院分区：
文献类型：
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作者：
J. Baker-Jarvis;M. Racine;Jihad Alameddine
利用最大熵最小范数法，给出了求解微分方程的一般方法。该技术是Baker - Jarvis [J]先前工作的推广和扩展。数学。物理学报，30,302（1989）。我们发现，在解向量的范数上引入一个额外的约束会产生一个在整个实轴上可积的概率分布。出现了许多简化。在这种扩展方法中，拉格朗日乘子和解向量可以显式地求解，从而消除了像以前的方法那样求解非线性方程组的拉格朗日乘子的必要性。结果表明，所得到的解等价于最小范数近似。具有傅里叶矩的微分方程的最大熵解被证明与傅里叶级数解相同。此外，还将该方法应用于求解随机漫步方程和Fokker-Planck方程。
The method of maximum entropy–minimum norm is utilized to produce a general method of solving differential equations. The technique is a generalization and extension of previous work performed by Baker‐Jarvis [J. Math. Phys. 30, 302 (1989)]. It is found that introducing an additional constraint on the norm of the solution vector produces a probability distribution that is integrable over the entire real axis. A number of simplifications occur. In this extended method the Lagrange multipliers and solution vector can be solved for explicitly, thus eliminating the necessity of solving systems of nonlinear equations for the Lagrange multipliers, as was required in the previous approach. It is shown that the solution obtained is equivalent to a minimum norm approximation. The maximum entropy solution of differential equations with Fourier moments is shown to be identical to a Fourier series solution. Additionally, the new method is applied to solving the random walk and Fokker–Planck equations.