Similarity solutions for nonlinear diffusion — a new integration procedure

✍ Scribed by J. M. Hill

• Publisher: Springer
• Year: 1989
• Tongue: English
• Weight: 627 KB
• Volume: 23
• Category: Article
• ISSN: 0022-0833
• DOI: 10.1007/bf00128865

✦ Synopsis

The nonlinear diffusion equation arises in many important areas of science and technology and most of the known exact solutions turn out to be similarity solutions. For a general similarity solution involving an arbitrary parameter 1, a new integration procedure is proposed which enables first integrals to be obtained for special values of A. The best known exact solutions arise from this analysis when the integration constant is taken to be zero and the procedure provides a natural way of deducing other special exact solutions. A new exact solution is obtained for the power law diffusivity of index -4/3 and new first integrals are deduced for a general equation which includes nonlinear cylindrical and spherical symmetrical diffusion and one-dimensional nonlinear diffusion with an inhomogeneous diffusivity. The procedure has given rise to an extensive number of first-order ordinary differential equations which include a wide variety of differing physical situations and which warrant further study either analytically to determine exact integrals or numerically for particular boundary value problems.