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Question 3 a. Using the method of separation of variables, show that the solution to the two-dimensional Laplace's equation 0≤ ≤5, 0≤ y ≤ 1, Fu Pu + ar² dy² subject to the boundary conditions u(0, y) = 0, u(5, y) = 0, can be written as = 0, u(z,0) = 0, u(x, 1) = f(x), an infinity series, u(z,y) = Asin("57 )sinh("57). n=1 Derive the integral expressions for the coefficients involving the bound- ary condition f(z). (You may assume without proof, that the functions sin(- naz 5) are orthogonal on the interval [0, 5]). b. Given the specific boundary condition f(x)= x(5-1), find the appro- priate coefficients An, n = 1,2,3,.....

Question 3 a. Using the method of separation of variables, show that the solution to the two-dimensional Laplace's equation 0≤ ≤5, 0≤ y ≤ 1, Fu Pu + ar² dy² subject to the boundary conditions u(0, y) = 0, u(5, y) = 0, can be written as = 0, u(z,0) = 0, u(x, 1) = f(x), an infinity series, u(z,y) = Asin("57 )sinh("57). n=1 Derive the integral expressions for the coefficients involving the bound- ary condition f(z). (You may assume without proof, that the functions sin(- naz 5) are orthogonal on the interval [0, 5]). b. Given the specific boundary condition f(x)= x(5-1), find the appro- priate coefficients An, n = 1,2,3,.....

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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please do a) and b)

Question 3 a. Using the method of separation of variables, show that the solution to the two-dimensional Laplace's equation Fu Fu + ər² dy² subject to the boundary conditions u(0, y) = 0, u(5, y) = 0, = 0, can be written as 0≤I≤5, 0≤ y ≤ 1, u(z,0)=0, u(r, 1) = f(x), an infinity series, u(z,y) =ỪAsin("57 )sinh("). Derive the integral expressions for the coefficients involving the bound- ary condition f(z). (You may assume without proof, that the functions sin(- naz 5) are orthogonal on the interval [0, 5]). b. Given the specific boundary condition f(x)= x(5-2), find the appro- priate coefficients An, n = 1,2,3,....
Transcribed Image Text:Question 3 a. Using the method of separation of variables, show that the solution to the two-dimensional Laplace's equation Fu Fu + ər² dy² subject to the boundary conditions u(0, y) = 0, u(5, y) = 0, = 0, can be written as 0≤I≤5, 0≤ y ≤ 1, u(z,0)=0, u(r, 1) = f(x), an infinity series, u(z,y) =ỪAsin("57 )sinh("). Derive the integral expressions for the coefficients involving the bound- ary condition f(z). (You may assume without proof, that the functions sin(- naz 5) are orthogonal on the interval [0, 5]). b. Given the specific boundary condition f(x)= x(5-2), find the appro- priate coefficients An, n = 1,2,3,....
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