Heat conduction and a general class of polynomials (Q2717873)

A general class of polynomials for finding solutions of the heat conduction equation [see \textit{B. R. Bhonsle}, Math. Jap. 11, 83-90 (1966)] NEWLINE\[NEWLINE\frac{\partial V}{\partial t}=k\frac{\partial^2V}{\partial x^2}-kx^2VNEWLINE\]NEWLINE is considered. These polynomials are obtained assuming that NEWLINE\[NEWLINEV(x,t)= \sum_{j=0}^\infty a_je^{-(1+2j)kt}H_j(x)NEWLINE\]NEWLINE and NEWLINE\[NEWLINEV(x,0)=x^\alpha \sum_{j=0}^{[n/m]}\frac{(-n)_{mj}}{j!}A_{n,j}\delta^jx^{2pj},NEWLINE\]NEWLINE where \(H_j(x)\) denotes the Hermite polynomial. Particular cases of \(A_{n,j}\) are considered. Utilization of polynomials in the heat conduction problem in a finite bar NEWLINE\[NEWLINE\frac{\partial V}{\partial t}=k\frac{\partial}{\partial x} \left((1-x^2)\frac{\partial V}{\partial x}\right), \quad x\in(-1,1),\;t>0; \quad \frac{\partial V}{\partial x}(\pm 1,t)=0NEWLINE\]NEWLINE and in the heat production problem in a cylinder NEWLINE\[NEWLINE\frac{\partial V}{\partial t}=\frac{k}{r} \frac{\partial}{\partial r}\left(r\frac{\partial V}{\partial r}\right)+ Kf(r)g(t); \quad V(a,t)=0, \quad V(r,0)=0NEWLINE\]NEWLINE are also investigated.