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中文摘要：
研究了在数学、力学中广泛出现的一类三阶非线性强阻尼发展扰动偏微分方程，并求其近似解析解．首先，构造一个泛函同伦映射，将方程的解表示以人工参数的幂级数形式，代人同伦映射，得到一个非线性扰动方程解的逐次迭代关系式，并考虑对应的一个无扰动项情形下的强阻尼发展方程，利用Fourier变换理论，求出其精确解．其次，以得到的精确解为同伦映射迭代式的初始函数，通过非线性扰动方程解的迭代关系式，再用Fourier变换法求解对应的方程．最后，便依次地得到了非线性强阻尼发展扰动偏微分方程的各次近似解析解．用上述方法得到的各次近似解，具有便于求解、精度高等特点．
英文摘要：
Widely emerging in the fields of mathematics and mechanics, a class of 3rd-order nonlinear strong-damp disturbed partial differential evolution equations were studied. Firstly, a functional homotopic mapping was constructed to express the solution to the evolution equation in a form of power series with artificial parameters, which was substituted into the homotopic mapping to build a method of successive iteration for the solution to the nonlinear disturbed equation. Then the corresponding non-disturbed strong-damp evolution equation was analyzed with exact solution based on the theory of Fourier transform. Secondly, the found exact solution was used as the initial function of the homotopic mapping iteration, and the iteration expansion of the nonlinear disturbed equation was applied to solve the related equations with the Fourier transform method. Finally, both the exact and arbitrary-order approximate analytic solutions to the nonlinear strong-damp disturbed evolution equation were obtained. The proposed homotopic mapping method is proved to have the advantages of convenience and accuracy.
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