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)