Mathematical analysis and numerical computation of nonlinear partial differential equations

冈本发现了Navier-Stokes方程的一些新的精确解，从而揭示了方程的奇异摄动分析。其中之一是Tamada和Dorrepaal的滞滞点流的推广，滞滞点流斜向壁上收敛。其他解包括满足Leray相似方程的解。其中，他发现了用超几何函数表示的解。他还研究了满足流入/流出边界条件的一类稳态Navier-Stokes方程的稳定性。有些解在所有雷诺数下都是稳定的，这是一个令人惊讶的结果。Okamoto和Sakajo用涡旋法对剪切流中的涡旋片进行了数值计算。t . kawai和Y.Takei通过代数分析方法研究了Painleve方程奇异摄动理论的某些方面。他们的理论将发表在Iwanami shotenk . murota和S.Iwata合著的《奇异摄动的代数分析》（日文）一书中，他们通过数值线性代数理论对目前的研究做出了贡献。某些奇异摄动问题经过适当的离散化后，得到一个非常病态的矩阵问题。他们所开发的技术对解决这类病态矩阵问题很有帮助。Murota提出了许多提高计算精度和速度的方法。

H.Okamoto discovered some new exact solutions of the Navier-Stokes equations which shed light on the singular perturbation analysis of the equations. One of them is a generalization of stagnation-point flows of Tamada and Dorrepaal which converges on a wall obliquely. Other solutions include those solutions which satisfy Leray's similarity equations. Among others, he found those solutions which are represented by the confluent hyper-geometric functions. He also studied the stability of certain stationary Navier-Stokes equations which satisfy the inflow / outflow boundary condition. Some the solutions are proved to be stable for all the Reynolds number, which is a surprising results. By the vortex method, Okamoto and Sakajo computed numerically the vortex sheet in shear flows. Some interesting interactions between vortex sheet and shear are discovered.T.Kawai and Y.Takei studied, via the algebraic analysis method, a certain aspect of singular perturbation theory of the Painleve equations. Their theory will be published in their book entitled "Algebraic Analysis of Singular Perturbation" (in Japanese) from Iwanami Shoten.K.Murota and S.Iwata made a contribution to the present study through the theory of numerical linear algebra. Some singular perturbation problem leads to a very ill-conditioned matrix problem after a suitable discretization. The techniques developed by them are very helpful when we solve such ill-conditioned matrix problem. Murota proposes many methods which leads to better accuracy and speed-up of the computation.