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5.20 Similarity variables for the heat equation: the purpose...

5.20 Similarity variables for the heat equation: the purpose of this exercise is to derive an important canonical solution for the heat equation and to introduce the method of similarity variables. (a) Consider the heat equation ut - uxx=0 - = X € R, t > 0. (5.102) Set u(x,t)=0(A(...

1. Calculate the Fourier transform of f, then use Theorem 6....

1. Calculate the Fourier transform of f, then use Theorem 6.2 to describe the function 1 8 8 g(x) = eiax f(2)e-iaz dzda. 27 - -00 -00 b) f(x) = e2c , if I < e-z . if x > 0 4. Find a function which is its own Fourier transform, i.e., for which Flf(x) = f(a). 9. Prove th...

-7 III Laplace/Poisson equation. 5. It is known from the t...

-7 III Laplace/Poisson equation. 5. It is known from the theory of electromagnetism that the electrostatic potential 4 ( (7) = satisfies the Poisson equation : (19 11 - 1/20 3 (x), where 3 (x) is the density of electric charges and Eo is a physical constant. do, IX/RR Let , XI>R ,...

2.1 For each of the following differential equations determi...

2.1 For each of the following differential equations determine the equi- librium solutions and examine their stability (b) du -2(1+u²)-1arctan ( = - - u dt 2.4 Consider the differential equation du dt = r ( 1 - u ) - u-mu, r, Investigate the stability of equilibrium solutions as u pas...

(1) Establish that L{th} = s-n-1n! and C{f(t -a)H(t -a)} = e...

(1) Establish that L{th} = s-n-1n! and C{f(t -a)H(t -a)} = e-asf(s).Use = a Laplace transform to solve xaut + Ux for x > 0, subject to u(x, 0) = 0 and u(0,t) = 0. Consider all possible choices of the (finite) parameter a > 0. (2) Establish that C{8(x - = for U > 0. Use a La...

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