3.1 Partial Differential Equations in Physics and Engineering 49 3.3 Solution of the One Dimensional Wave Equation: The Method of Separation of Variables 52 3.4 D’Alembert’s Method 60 3.5 The One Dimensional Heat Equation 69 3.6 Heat Conduction in Bars: Varying the Boundary Conditions 74 3.7 The Two Dimensional Wave and Heat Equations 87

Separation of Variables is a special method to solve some Differential Equations A Differential Equation is an equation with a function and one or more of its derivatives : Example: an equation with the function y and its derivative dy dx

4. Be able to solve the equations modeling the heated bar using Fourier’s method of separation of variables 25.2 Introduction When a function depends on more than one variable it has partial derivatives in-stead of ordinary derivatives. For 18.03 this means we will have to consider partial differential equations (PDE) involving such functions.

The one-dimensional wave equation Separation of variables The two-dimensional wave equation Solution by separation of variables We look for a solution u(x,t)intheformu(x,t)=F(x)G(t). Substitution into the one-dimensional wave equation gives 1 c2 G(t) d2G dt2 = 1 F d2F dx2. Since the left-hand side is a function of t only and the Based on the modified homotopy perturbation method (MHPM), exact solutions of certain partial differential equations are constructed by separation of variables and choosing the finite terms of a ...

Separation of Variables is a special method to solve some Differential Equations A Differential Equation is an equation with a function and one or more of its derivatives : Example: an equation with the function y and its derivative dy dx