任意变系数微分方程的精确解析法
Exact Analytic Method for Solving Variable Coefficient Differential Equation
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摘要: 工程中的许多问题归结为求解任意变系数微分方程的解.本文首次提出精确解析法,用以求解任意变系数微分方程在任意边界条件下的解.文中还给出精确解析法的一般计算格式,得到了一致收敛于精确解及其任意阶导数的解析表达式,并给出收敛性证明.文末给出四个算例,均得到较好的结果,证明了本文理论的正确性.Abstract: Many engineering problems can be reduced to the solution of a variable coefficient differential equation.In this paper,the exact analytic method is suggested to solve variable coefficient differential equations under arbitrary boundary condition.By this method,the general computation formal is obtained.Its convergence in proved.We can get analytic expressions which converge to exact solution and its higher order derivatives uniformy Four numerical examples are given,which indicate that satisfactory results can he obtaned by this method.
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[1] 叶开沉,非均匀变厚度弹性力学的若干问题的一般解,N.非均匀变厚度梁的弯曲、稳定性和自由振动,兰州大学学报力学专号,1(1979),133-157. [2] Rektorys.Karel,Variational Methods in Mathematics,Science and Engineering.sec ed.,D.Reidel Publishing Company.Holland(1980).328-336. [3] 纳依玛克,M.A.,《线性微分算子》,科学出版社,北京(1964);13-28. -
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