### Similarity solutions of nonlinear partial differential equations

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Chapter 1 gives an introduction to the text.

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The author describes research work in the area and describes the types of problems where nonlinear differential equations are found. The search for exact solutions is motivated by the desire to understand the mathematical structure of the solutions themselves and to obtain a deeper understanding of the physical phenomena being described.

The idea is to illustrate and bring out main points of the problems included. Most examples are drawn from real physical situations, mainly from fluid mechanics and nonlinear diffusion. Chapter 2 gives an interesting discussion of first order, linear partial differential equations. Quasilinear partial differential equations of first order are described as are initial value problems.

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Fourteen example problems are solved. Chapter 3 is on exact similarity solutions of nonlinear partial differential equations, presented in two parts. Part A is on the reduction of partial differential equations by infinitesimal transformations. Systems of partial differential equations are described. Part B discusses a nonlinear heat equation in three dimensions.

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Also discussed is the similarity solution of Burgers equation elementary nonlinear diffusive traveling wave equation by the direct method. Exact solutions of equations for free surface flows for shallow water are solved by the direct similarity approach.

## On the derivation of first integrals for similarity solutions

An example from gas dynamics is solved by the multipronged approach to finding exact solutions of nonlinear partial differential equations. Chapter 4 is about traveling wave problems.

The chapter begins with a description of the problem and the general wave equation. The one-dimensional wave equation in gas dynamics is given and a solution is provided.

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- Similarity Solutions Nonlinear Partial Differential by Dresner Lawrence.

The elementary nonlinear diffusive traveling wave equation is described next, followed by equations for traveling waves for higher-order diffusive systems. Multidimensional systems of homogeneous partial differential equations written to describe simple wave flows are discussed and solved.

## Search Results for Initial value problems -- Numerical solutions

Traveling wave equations for nonhomogeneous hyperbolic or dispersive systems are written and solved, as are equations for hydromagnetic traveling waves. Simple waves on shear flows are also discussed. Chapter 5 is on the exact linearization of nonlinear partial differential equations. Burgers equation in one and higher dimensions is written and analyzed.

The nonlinear degenerate diffusion equation is also written and a numerical solution is described. An equation for the one-dimensional motion of an ideal compressible isentropic gas and the Born-Infeld equation are discussed. Another interesting example involves the mathematical description of water waves on a uniformly sloping beach. The chapter concludes with a description of simple waves on shear flows. Similarity solutions and transformations are given in a most general form derived to the first time in terms of reciprocal Jacobian elliptic functions.

By using a special transformation the first derivative of the equation can be transformed off leading to a further nPDE.

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The latter equation is also studied as well as algebraic properties and group invariant solutions could be derived. This new classes of solutions obtained are closely related to solutions of the kinetic model and so far, expressions for a generating function considering normalized moments are also deduced. For practical use in numerical calculations some series representations are given explicitly. This approach presents similarity solutions in an entirely new light and by equal utilization of both the partial differential equation to be solved and the first order partial differential equation we are able to deduce previous integrals virtually immediately without 'reference to the underlying ordinary differential equation.

Amiya K. Pani and Robert S.

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Anderssen, eds. Proceedings of the Centre for Mathematics and its Applications, v. Dates First available in Project Euclid: 18 November Zentralblatt MATH identifier