A review is made on recent developments of exact methods for solving soliton equations. Among various exact methods known nowadays, we mainly survey the bilinear transformation method (BTM) initiated by R. Hirota. The essence of the BTM is illustrated concretely by applying it to the KdV equation which is a typical soliton equation.
The relations of the BTM to other exact methods such as the inverse scattering method, the Backlund transformation and the Sato theory (the theory of τ function) are also discussed. Finally, some linearizable nonlinear equations are considered on the basis of the BTM.

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A review is given on the basic idea of the method of renormalized expansions and on the properties of Eulerian and Lagrangian approximations of turbulence derived by the method.

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The analytical methods by which nonlinear interfacial fluid motions can be modelled are introduced. The partial differential equations which completely satisfy the interfacial conditions or the free surface conditions are derived. In particular, the model equation of internal waves involves two indefinite functions and these are related to the difference of velocities and the difference of pressure gradients at the interface respectively.
Furthermore, for irrotational motions the indefinite function can be determined so as to satisfy other boundary conditions such as bottom conditions. Finally, the modelling of the Rayleigh-Taylor problem and the breaking waves in the finite depth channel are discussed by the present method.

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The unsaturated infiltration through soil was numerically simulated by a Random Walk Method. Several cases of the time development of water content in a soil and the movements of water-particles when there was a constant rate rainfall for a certain duration of time were computed. The infiltrations through a soil with a homogeneous infiltration coefficient and with non-homogeneous coefficients were computed, whose results were in good agreement with laboratory experimental data.

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Analytical solutions and numerical solutions for the incompressible flow of a viscous fluid past a sphere are presented for a range of Reynolds number, Re, from Re=0.01 to Re=120. Flow patterns, drag and pressure distributions as predicted by analytical solutions to the Oseen equations of motion are compared with ones by numerical solutions to the Navier-Stokes equations of motion. In the range of low Reynolds numbers, they agree better with the ones by numerical solutions than the analytical predictions by the Proudman-Pearson method. But at intermediate Reynolds numbers, the analytical solutions for most properties deviate progressively from the numerical ones withRe. However, it should be noted that the qualitative nature of the flow field can be described by the Oseen solutions even at intermediate Reynolds numbers.
Lastly, successive approximations to the Navier-Stokes equations are carried out with the Oseen solutions ; the results for the first approximations up to Re=10 are useful for numerical solutions of the Navier-Stokes equations.

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A numerical method for computing inviscid transonic flow around an arbitrary aircraft configuration is described in this paper. Basic equations are simultaneous equations for velocity potential and Mach number. These equations are equivalent to the full potential equation. The simultaneous equations are numerically solved satisfying given boundary conditions. Shock waves are captured as discontinuous surfaces where shock waves are expected to occur.
The calculated flow field is a rectangular box, where grid points and boundary points are generated in Cartesian coordinates. The grid points are equidistantly distributed, but their intervals can be varied with the location. External forms of the aircraft and lifting surfaces are shaped by use of numerous small triangular planes. The majority of the boundary points are on these external forms. The basic equations are expressed as difference equations using the grid points and the boundary points. The difference equations are solved in the numerical procedure. Several calculated results of this method are compared with the experimental and other numerical results.

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