Chapter2-Book - Chapter 2 Method of Separation of Variables...

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Chapter 2 Method of Separation of Variables 2.1 Introduction In Chapter 1 we developed from physical principles an understanding of the heat equation and its corresponding initial and boundary conditions. We are ready to pursue the mathematical solution of some typical problems involving partial differential equations. We will use a technique called the method of separation of variables. You will have to become an expert in this method, and so we will discuss quite a few examples. We will emphasize problem-solving techniques, but we must also understand how not to misuse the technique. A relatively simple but typical, problem for the equation of heat conduction occurs for a one-dimensional rod (0 x L ) when all the thermal coefficients are constant. Then the PDE, ∂u ∂t = k 2 u ∂x 2 + Q ( x, t ) , t > 0 0 < x < L, (2.1.1) must be solved subject to the initial condition, u ( x, 0) = f ( x ) , 0 < x < L, (2.1.2) and two boundary conditions. For example, if both ends of the rod have prescribed temperature, then u (0 , t ) = T 1 ( t ) u ( L, t ) = T 2 ( t ) . t > 0 (2.1.3) The method of separation of variables is used when the partial differential equation and the boundary conditions are linear and homogeneous, concepts we now explain. 35
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36 Chapter 2. Method of Separation of Variables 2.2 Linearity As in the study of ordinary differential equations, the concept of linearity will be very important for us. A linear operator L by definition satisfies L ( c 1 u 1 + c 2 u 2 ) = c 1 L ( u 1 ) + c 2 L ( u 2 ) (2.2.1) for any two functions u 1 and u 2 , where c 1 and c 2 are arbitrary constants. ∂/∂t and 2 /∂x 2 are examples of linear operators since they satisfy (2.2.1): ∂t ( c 1 u 1 + c 2 u 2 ) = c 1 ∂u 1 ∂t + c 2 ∂u 2 ∂t 2 ∂x 2 ( c 1 u 1 + c 2 u 2 ) = c 1 2 u 1 ∂x 2 + c 2 2 u 2 ∂x 2 . It can be shown (see Exercise 2.2.1) that any linear combination of linear operators is a linear operator. Thus, the heat operator ∂t k 2 ∂x 2 is also a linear operator. A linear equation for u is of the form L ( u ) = f, (2.2.2) where L is a linear operator and f is known. Examples of linear partial differential equations are ∂u ∂t = k 2 u ∂x 2 + f ( x, t ) (2.2.3) ∂u ∂t = k 2 u ∂x 2 + α ( x, t ) u + f ( x, t ) (2.2.4) 2 u ∂x 2 + 2 u ∂y 2 = 0 (2.2.5) ∂u ∂t = 3 u ∂x 3 + α ( x, t ) u. (2.2.6) Examples of nonlinear partial differential equations are ∂u ∂t = k 2 u ∂x 2 + α ( x, t ) u 4 (2.2.7) ∂u ∂t + u ∂u ∂x = 3 u ∂x 3 . (2.2.8) The u 4 and u∂u/∂x terms are nonlinear; they do not satisfy (2.2.1).
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2.2. Linearity 37 If f = 0, then (2.2.2) becomes L ( u ) = 0, called a linear homogeneous equa- tion. Examples of linear homogeneous partial differential equations include the heat equation, ∂u ∂t k 2 u ∂x 2 = 0 , (2.2.9) as well as (2.2.5) and (2.2.6). From (2.2.1) it follows that L (0) = 0 (let c 1 = c 2 = 0). Therefore, u = 0 is always a solution of a linear homogeneous equation. For example, u = 0 satisfies the heat equation (2.2.9). We call u = 0 the trivial solution of a linear homogeneous equation. The simplest way to test whether an equation is homogeneous is to substitute the function u identically equal to zero. If u 0
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