A equação de Bessel e o problema da difusão de calor num cilindro infinito.

Abstract

Power series and of great importance in solving differential equations, with results that can be used as a basis both for the representation of functions, mainly special functions, and for application in various types of problems. This work is related to the process of heat diffusion through an in nite cylinder, modeled by a partial differential equation. Such an equation, called the diffusion equation The resolution established here proposes to solve this equation in its particular formulation for the case of an infinite cylinder with racial distribution and prescribed temperature. Particularly given via separation method in the following form t = # exp[ ak2 ]; where # is the solution of the differential equation r2# + k2# = 0: In our case #(x) is a solution of the Bessel equation, represents by means of power series, around an ordinary point or a singular point regular. Initially, aiming at the application of the Frobenius method, an extension of the power series method, conditions are established for the ordinary point to be characterized as a removable regular singular point, in such a way that the Frobenius method also produces the so-called analytic solutions around an ordinary point, possibly multiplying the series by a logarithmic term or by a fractional power exponent. Then, using its respective recurrence relation, the Euler-Cauchy equation, two solutions of the ODE are determined. The analysis of the linear dependence or independence of such solutions is done through the Theorem of the General Solution of the Bessel Equation.