Some General Methods of Solution Applied to Differential Equations Arising from Problems in Mathematical Physics

Abstract

(Page numbers are given in brackets after sub-headings.) Chapter 1 :-Application of operational methods of solution to ordinary differential equations with constant coefficients. Motion of helical springs --- Free vibration ( p.2 ) --- Forced vibration ( p.7 ) --- Resonance ( p.10 ) ----- Linked springs ( p.12 ) --- Analogous electrical problems ( P. 15 ). Appendix A :-Laplace Transformations. ( p.18 ) Appendix B :- Brief explanation of the operational methods of solution used in the chapter. ( p.19 ) Appendix C :- Example of partial fraction working. ( p.21a) Chapter 2 :- Application of Fourier Series and of operational methods of solution to partial differential equations. Small transverse vibrations of a light elastic string ( p.22 ) Vibration of a string initially plucked at its middle point ( p.28 ) --- Alternative method of solution of the preceeding problem by operational methods ( p.31 ) --- Discussion of the shape assumed by the string at any time ( p.36 ) Chapter 3 :- Application of Bessel Functions to the solution of partial differential equations of the second order. Small vibrations of a light circular membrane ( p.39 ) Appendix A :- Transformation of the differential equation of motion from Cartesian to cylindrical coordinates. ( p.46 ). Appendix B :- Note on roots of Bessel Functions. ( p.47 ) Appendix C :- Bessel Functions - Some important results applied in the chapter. p.48 ). Chapter 4 :- Application of Legendre Polynomials to the solution of partial differential equations of the second order. Conduction of heat in a solid body ( p.50 ) ---------- Flow of heat in a solid sphere ( p.53 ). Appendix :-Statement of the Divergence Theorem used in the chapter. ( p.61 ). Chapter 5 :- Example of Simultaneous Partial Differential Equations. The flow of electricity in a long imperfectly insulated cable ( p.62 ) ---- Analogous problem of the flow of heat in rod of small uniform cross section ( p.69 ).