You need to solve only (3)part and solve plz perfectly i need perfect soloution take your time and give me 100 percent exact soloution. Plz Separation of variablesThe lateral radiation of heat from a bar of length L insulted at both ends can be modelled withthe linear partial differential equation (PDE)Ou8²u<= D -au for 00,02²(1)Otwith the conditionsu, (0.1)=₂(L,t)=0 and u(,0) = f(x).where Lo, DER are positive constants.1. Seek a separation solution of the form u(x, t) = X(z)T(t) to showT 0DTD X= k₁where & denotes the separation constant.] 2. Use equation (2) to derive two ordinary differential equations (ODEs), one in spacer andone in time t.3. Determine the boundary conditions for the ODE that depends on z.4. Find the non-trivial solutions of X(r)= X(r) and corresponding values of the separationconstant k=k, for n= 0, 1, 2, 3.....Hint: you may need to consider cach of the three causes k=-p², k = 0, k=p² (p0) tofind all the non-trivial solutions5. Find the solutions T(t)= T(t), for n=0,1,2,3....6. Use the superposition principle for linear PDEs to write down the solution for u that satisfiesthe boundary conditions (0,t)=u₂(L,t) = 0.7. Determine expressions for the unknown coefficients in your solution for that satisfies theinitial condition u(,0) = f(x).8. Find the solution when f(x)=1+ cos rr, for values of L=1.0=1 and D = 1.¹9. In a single figure, plot the solution found in Question 8 at times t=0,0.25, 0.5, 2. Brieflydescribe the behaviour as t increases. What is the constant value temperature of theenvironment that surrounds the bar?19

There are so many subparts, as your requirement i solve first three subparts, including part 3.…

Separation of variables The lateral radiation of heat from a bar of length L insulted at both ends can be modelled with the linear partial differential equation (PDE) du Pu = D -au for 00, 02² (1) Ət with the conditions ₂ (0,1)=₂(L,t)=0 and u(z,0)=f(x). where L.a, DER are positive constants. 1. Seek a separation solution of the form u(z,t) = X(z)T(t) to show +==k where & denotes the separation constant. ] 2. Use equation (2) to derive two ordinary differential equations (ODEs), one in spacer and one in time f. 3. Determine the boundary conditions for the ODE that depends on z.

Separation of variables The lateral radiation of heat from a bar of length L insulted at both ends can be modelled with the linear partial differential equation (PDE) du Pu = D -au for 00, 02² (1) Ət with the conditions ₂ (0,1)=₂(L,t)=0 and u(z,0)=f(x). where L.a, DER are positive constants. 1. Seek a separation solution of the form u(z,t) = X(z)T(t) to show +==k where & denotes the separation constant. ] 2. Use equation (2) to derive two ordinary differential equations (ODEs), one in spacer and one in time f. 3. Determine the boundary conditions for the ODE that depends on z.

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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