Axisymmetric Problems in Cylindrical Coordinates

Abstract

Problems of fractional thermoelasticty based on the time-fractional heat conduction equation are considered in cylindrical coordinates. Heat conduction in a long cylinder, in an infinite solid with a long cylindrical cavity and in a half-space is investigated. The solutions to the fractional heat conduction equation under the Dirichlet boundary condition with the prescribed boundary value of temperature and the physical Neumann boundary condition with the prescribed boundary value of the heat flux are obtained. Different kinds of integral transforms (Laplace, Fourier and Hankel) are used. Associated thermal stresses are investigated. The representations of stresses in terms of the displacement potential and the biharmonic Love function allows us to satisfy the prescribed boundary condition for the stress tensor.

In order to solve a differential equation you

look at it till a solution occurs to you. George Pólya

Appendix: Integrals

Integrals containing elementary functions are taken from [17], integrals (4.216) and (4.217) containing Bessel function are taken from [18], integrals (4.212) and (4.214) are presented in [1]; integrals (4.202), (4.213) and (4.215) have been evaluated by the author.

$$\begin{aligned}&\int _{0}^{\infty } \frac{1}{x^{2}+ c^{2}}\, \text {e}^{-a^{2}x^{2}} \text {d}x = \frac{\pi }{2c} \, \text {e}^{a^{2}x^{2}} \text {erfc}\,(ac) \end{aligned}$$

(4.201)

$$\begin{aligned}&\int _{0}^{\infty } \frac{1}{(x^{2}+ c^{2})^{2}}\, \text {e}^{-a^{2}x^{2}} \text {d}x = \frac{\pi }{4c^{3}} \, \text {e}^{a^{2}c^{2}} \bigg [ \left( 1-2a^{2}c^{2} \right) \text {erfc} \left( ac \right) \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad + \frac{2ac}{\sqrt{\pi }}\, \exp \left( -a^{2}c^{2} \right) \bigg ] \end{aligned}$$

(4.202)

$$\begin{aligned}&\int _{0}^{\infty } \frac{x}{x^{2}+ c^{2}}\, \sin (bx)\, \text {d}x = \frac{\pi }{2} \, \text {e}^{- bc} \end{aligned}$$

(4.203)

$$\begin{aligned}&\int _{0}^{\infty } \frac{ 1}{x(x^{2}+c^{2})} \, \sin (bx) \, \text {d}x = \frac{\pi }{2c^{2}} \left( 1 - \text {e}^{- bc} \right) \end{aligned}$$

(4.204)

$$\begin{aligned}&\int _{0}^{\infty } \frac{x}{(x^{2}+ c^{2})^{2}}\, \sin (bx)\, \text {d}x = \frac{\pi b}{4c} \, \text {e}^{- bc} \end{aligned}$$

(4.205)

$$\begin{aligned}&\int _{0}^{\infty } \frac{1}{x^{2}+ c^{2}}\, \cos (bx)\, \text {d}x = \frac{\pi }{2c} \, \text {e}^{-bc} \end{aligned}$$

(4.206)

$$\begin{aligned}&\int _{0}^{\infty } \frac{1}{(x^{2}+ c^{2})^{2}}\, \cos (bx)\, \text {d}x = \frac{\pi (1+bc)}{4c^{3}} \, \text {e}^{-bc} \end{aligned}$$

(4.207)

$$\begin{aligned}&\int _{0}^{\infty } x \, \text {e}^{-a^{2}x^{2}}\, \sin (bx) \, \text {d}x = \frac{\sqrt{\pi }b}{4a^{3}} \ \exp \left( - \frac{b^{2}}{4a^{2}} \right) \end{aligned}$$

(4.208)

$$\begin{aligned}&\int _{0}^{\infty } \frac{1}{x} \, \text {e}^{-a^{2}x^{2}}\, \sin (bx )\, \text {d}x = \frac{\pi }{2} \ \text {erf} \left( \frac{b}{2a} \right) \end{aligned}$$

(4.209)

$$\begin{aligned}&\int _{0}^{\infty } \text {e}^{- a^{2}x^{2}}\, \cos (bx) \, \text {d}x = \frac{\sqrt{\pi }}{2a} \ \exp \left( - \frac{b^{2}}{4a^{2}} \right) \end{aligned}$$

(4.210)

$$\begin{aligned}&\int _{0}^{\infty } x^{2} \, \text {e}^{-a ^{2}x^{2}} \cos ( bx) \, \text {d}x = \frac{\sqrt{\pi }}{4a^{3}} \left( 1- \frac{b^{2}}{2a^{2}} \right) \, \exp \left( - \frac{b^{2}}{4a^{2}} \right) \end{aligned}$$

(4.211)

$$\begin{aligned}&\int _{0}^{\infty } \frac{x}{x^{2}+ c^{2}}\, \text {e}^{-a^{2}x^{2}}\, \sin bx\, \text {d}x = \frac{\pi }{4} \, \text {e}^{a^{2}c^{2}} \bigg [ \text {e}^{-bc} \text {erfc} \left( ac - \frac{b}{2a} \right) \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \quad - \, \text {e}^{bc} \text {erfc} \left( ac + \frac{b}{2a} \right) \bigg ] \end{aligned}$$

(4.212)

$$\begin{aligned}&\int _{0}^{\infty } \frac{x}{(x^{2}+ c^{2})^{2}}\, \text {e}^{-a^{2}x^{2}}\, \sin \left( bx \right) \, \text {d}x = \frac{\pi }{8c} \, \text {e}^{a^{2}c^{2}} \bigg [ \left( b-2a^{2}c \right) \text {e}^{-bc} \text {erfc} \left( ac - \frac{b}{2a} \right) \nonumber \\&\qquad \qquad +\left( b+2a^{2}c \right) \text {e}^{bc} \text {erfc} \left( ac + \frac{b}{2a} \right) \bigg ] \end{aligned}$$

(4.213)

$$\begin{aligned}&\int _{0}^{\infty } \frac{1}{x^{2}+ c^{2}}\, \text {e}^{-a^{2}x^{2}}\, \cos bx\, \text {d}x = \frac{\pi }{4c} \, \text {e}^{a^{2}c^{2}} \bigg [ \text {e}^{-bc} \text {erfc} \left( ac - \frac{b}{2a} \right) \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \quad + \, \text {e}^{bc} \text {erfc} \left( ac + \frac{b}{2a} \right) \bigg ] \end{aligned}$$

(4.214)

$$\begin{aligned}&\int _{0}^{\infty } \frac{1}{(x^{2}+ c^{2})^{2}}\, \text {e}^{-a^{2}x^{2}}\, \cos \left( bx \right) \, \text {d}x = \frac{\pi }{8c^{3}} \, \text {e}^{a^{2}c^{2}} \bigg [ \frac{4ac}{\sqrt{\pi }}\, \exp \left( -a^{2}c^{2} - \frac{b^{2}}{4a^{2}} \right) \nonumber \\&\qquad \qquad + \left( 1+bc-2a^{2}c^{2} \right) \text {e}^{-bc} \text {erfc} \left( ac - \frac{b}{2a} \right) \nonumber \\&\qquad \qquad + \left( 1-bc-2a^{2}c^{2} \right) \text {e}^{bc} \text {erfc} \left( ac + \frac{b}{2a} \right) \bigg ] \end{aligned}$$

(4.215)

$$\begin{aligned}&\int _{0}^{a} x \, \sin \left( b\sqrt{a^{2}-x^{2}}\right) J_{0} (cx) \, \text {d}x \nonumber \\&\qquad \qquad = \frac{ab}{b^{2}+c^{2}} \left[ \frac{\sin \left( a\sqrt{b^{2}+c^{2}}\right) }{a\sqrt{b^{2}+c^{2}}} - \cos \left( a \sqrt{b^{2}+c^{2}}\right) \right] \end{aligned}$$

(4.216)

$$\begin{aligned}&\int _{0}^{a} \frac{x }{\sqrt{a^{2}-x^{2}}} \, \cos \left( b\sqrt{a^{2}-x^{2}}\right) \, J_{0} (cx) \, \text {d}x = \frac{\sin \left( a\sqrt{b^{2}+c^{2}}\right) }{\sqrt{b^{2}+c^{2}}} \end{aligned}$$

(4.217)