Generalized Trapezoidal Formulas For The Symmetric Heat Equation In Polar Coordinates II. The Case Of Point Sources

The present paper is in continuation of our previous paper Chawla et al. [4]. An important class of applications of the radially symmetric heat equation in polar coordinates: u_{t}=v ( u_{rr}\,+\; ( a/r ) u_{r} ), involve the presence of a continuous point source of heat at the centre of the sphere ( a=2 ) or on the axis of the cylinder ( a=1 ). This necessitates a modification of the radial grid used in [4]; our modification of the radial grid in the present paper accommodates a point source of heat at r=0 in a natural way. We then extend generalized trapezoidal formulas GTF( \alpha ) for the one-step time integration of these problems. Again, with the help of the generalized finite Hankel transforms introduced in [4] we are able to obtain, in a natural way, analytical solutions of the heat equation in the presence of point sources of heat for both the cases a=1 and a=2 . Numerical experiments are provided to compare the performance of the GTF( \alpha ) time integration scheme with the schemes based on the backward Euler and the classical arithmetic-mean trapezoidal formula.

Keywords:

• Symmetric Heat Equation
• Polar Coordinates
• Continuous Point Sources
• Generalized Trapezoidal Formulas
• Generalized Finite Hankel Transforms