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Homework4_Spring_2010 - where a is a constant(a Is the...

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1 MAE 105: Introduction to Mathematical Physics Homework 4 Posted: April 28, 2010 , DUE: May 7, 2010 (Friday, 1 pm), 1. A function f(x) has a period “ L ”, if f(x) = f(x+L) . Find the periods of the following functions, where n is an integer: (a) sin (n x ) , (b) e inx , (c) sin 2. Plot and then draw the odd and even extensions of the following functions, over a symmetric interval ( -L, L ). (a) f(x) = e -x , 0 < x <1, (b) f(x) = x 3 , 0 < x <1 3. Expand the function, f(x) = x , defined over the interval 0 < x <2 , in terms of: (a) A Fourier sine series, using an odd extension of f(x) (b) A Fourier cosine series, using an even extension of f(x) 4. Revision of the Method of Separation of Variables Any function of more than one variable, say g(x,y) , if it satisfies a linear homogeneous PDE can be solved by the method of separation of variables. Consider the PDE:
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Unformatted text preview: , where a is a constant. (a) Is the above PDE, (i) linear, and (ii) homogeneous? Why? (b) Express g(x, y) as the product of two individual functions X(x) and Y(y) . Solve for X(x) and Y(y) , through a similar procedure to that developed in class for the heat conduction equation. What is the final form of g(x, y) ? (c) We now solve for the constants of integration, in (b) through the use of boundary/initial conditions. Given that a boundary condition is g(x=0, y) = 8 e-3y + 4 e-5y , show that the complete solution for the PDE is g(x, y)= 8 e-3(a x + y ) + 4 e-5(a x + y ) Hint: Given that there are two terms in g(x=0, y) you have to use the Principle of Superposition to write down all the possible solutions....
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