Problem 1: Consider an infinite slab of unit width. The differential equation describing thesystem isd?TdTdT= 0dz? +8(z)dzdzz=0Iz=1g(z) = 10– 20zShow that the following function are eigenfunctions of the linear transformation:$,(z) = H, cos(@,z), where @, = (i – 1)7, H1 = 1, H;12. Determine thecorresponding eignenvalues.2) Using 4 eigenfunctions o, (z) = H, cos(@,z), where w, = (i– 1)7, H1 = 1, H¡ = /2, i =%3D%3D2...4, approximate the differential equation with a set of algebraic equations using theGalerkin method. Report your answer in the form of Ax =b by identifying x, A and band indicating how an approximation of T(z) can be constructed from x. You shouldfind the following identity usefulcos(@z), z sin(@ z)Szcos(@2)dz =3) Show that the solution to the system of equations from part 1 isT(z) = a, +(b, / n' )¢,(z)+(b, / 4n² )¢,(z)+(b, /9n² )¢,(z)%3Dwhere b; are element of the vector b and a, is any arbitrary number. Using MATLAB,plot T(z) with a, = 303.4) Repeat part 1 with g(z) = 10 and show that the resulting set of algebraic equations hasno solution. Discuss this conclusion in the context of Example 2.13 from the book(Graham and Rawlings).

NOTE : As per Bartleby rules, only first three parts have to be answered. Please upload others separately. Given that : At z=0 and z=1 , dTdz=0 Also by using the fact : 0=d2Tdz2+g(z) The value of g(z) is : g(z)=10-20z PART 1 The function is given as : ϕi(z)=Hicoswizϕi(z)=Hicosi-1πz The eigenvalues will be : H1=1H2=2H3=2+2iH4=2-2iPART 2 Using the above function and by parts, we get ∫z cos(wz) dz =cos(wz)w2+z sin(wz)w PART 3 By using part 1, The function for T is : T(z)=ai+b2π2ϕ2(z)+b24π2ϕ3(z)+b29π2ϕ4(z)

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Problem 1: Consider an infinite slab of unit width. The differential equation describing the system is d?T dT dT 0 = +g(z) = 0 = 0 dz? dz dz z=1 g(z) = 10 – 20z 1) Show that the following function are eigenfunctions of the linear transformation: $,(z) = H, cos(@,z), where @, = (i – 1)x , H1 1, Н. 12. Determine the corresponding eignenvalues. 2) Using 4 eigenfunctions , (z) = H, cos(@,z), where a, = (i– 1)T , H1 = 1, H¡ = /2, i 2...4, approximate the differential equation with a set of algebraic equations using the Galerkin method. Report your answer in the form of Ax = b by identifying x, A and b

Problem 1: Consider an infinite slab of unit width. The differential equation describing the system is d?T dT dT 0 = +g(z) = 0 = 0 dz? dz dz z=1 g(z) = 10 – 20z 1) Show that the following function are eigenfunctions of the linear transformation: $,(z) = H, cos(@,z), where @, = (i – 1)x , H1 1, Н. 12. Determine the corresponding eignenvalues. 2) Using 4 eigenfunctions , (z) = H, cos(@,z), where a, = (i– 1)T , H1 = 1, H¡ = /2, i 2...4, approximate the differential equation with a set of algebraic equations using the Galerkin method. Report your answer in the form of Ax = b by identifying x, A and b

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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Question

Problem 1: Consider an infinite slab of unit width. The differential equation describing the
system is
d?T
dT
dT
= 0
dz? +8(z)
dz
dz
z=0
Iz=1
g(z) = 10– 20z
Show that the following function are eigenfunctions of the linear transformation:
$,(z) = H, cos(@,z), where @, = (i – 1)7, H1 = 1, H;
12. Determine the
corresponding eignenvalues.
2) Using 4 eigenfunctions o, (z) = H, cos(@,z), where w, = (i– 1)7, H1 = 1, H¡ = /2, i =
%3D
%3D
2...4, approximate the differential equation with a set of algebraic equations using the
Galerkin method. Report your answer in the form of Ax =b by identifying x, A and b
and indicating how an approximation of T(z) can be constructed from x. You should
find the following identity useful
cos(@z), z sin(@ z)
Szcos(@2)dz =
3) Show that the solution to the system of equations from part 1 is
T(z) = a, +(b, / n' )¢,(z)+(b, / 4n² )¢,(z)+(b, /9n² )¢,(z)
%3D
where b; are element of the vector b and a, is any arbitrary number. Using MATLAB,
plot T(z) with a, = 303.
4) Repeat part 1 with g(z) = 10 and show that the resulting set of algebraic equations has
no solution. Discuss this conclusion in the context of Example 2.13 from the book
(Graham and Rawlings).

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