Partial Differential Equations on Bounded Domains
Part of the Undergraduate Texts in Mathematics book series (UTM)
This chapter treats a standard and important method called separation of variables, or the method of eigenfunction expansions, for solving partial differential equations on bounded spatial domains. This method, due to Fourier in the early 1800s, with contributions by Euler and D’Alembert in the late 1700s, is fundamental. In fact, many textbooks emphasize this method over all others because of its extensive applications in physics and engineering where most problems are on bounded spatial domains. In a nutshell, the essential feature of the method is the replacement of the partial differential by a set of ordinary differential equations which then are solved subject to given initial and boundary conditions. We make the assumption that u can be written as a product of a function of x and a function of t, that is,
If the method is to be successful, when this product is substituted into the PDE, the PDE separates into two ODEs, one for y(x) and one for g(t). Substitution of the product into the boundary conditions leads to boundary conditions on the function y(x). Therefore, we are faced with a spatial ODE boundary value problem for y(x) and a temporal ODE problem for g(t). When the equations for y(x) and g(t) are solved, we can form a product solution \(u(x,t)=y(x)g(t)\) of the PDE that satisfies the boundary conditions. The boundary value problem we obtain for y(x) is a special type of eigenvalue problem called a Sturm–Liouville problem and it has infinitely many solutions. Consequently, we will have infinitely many product solutions \(u_1(x,t),\, u_2(x,t),\, u_3(x,t),\ldots\) that satisfy the boundary conditions. By superimposing these solutions, or adding them up in a special way, we determine a solution of the PDE and boundary conditions that also satisfies the initial condition(s). In other words, we form the series
and choose the constants c n such that the sum satisfies the initial condition(s) as well. The result of the calculation is an infinite series representation of the solution to the the original initial boundary value problem for the PDE.
$$u(x,t)=c_1u_1(x,t)+c_2 u_2(x,t)+c_3 u_3(x,t) +\cdots$$
KeywordsHeat Equation Divergence Theorem Eigenfunction Expansion Liouville Problem Singular Problem
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