Solve the boundary-value problem for your own choice of ?...

Solve the boundary-value problem for your own choice of ? and f(x): ?²u/?x² + ?e?? = ?u/?t, 0 < x < ?, t > 0 u(0,t) = 0, ?u/?x(?,t) = 0, t > 0 u(x,0) = f(x), 0 < x < ? 1. What type of equation and boundary condition are these? Explain it in both mathematical and physical terms. 2. Find the "Steady-State" solution? 3. Find the "Transient" solution? 4. Write down the final formal solution of the entire BVP.

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00:01 In this question, we have been given a boundary value problem, which is del square u over del x square plus alpha e raised to minus x equals to del u over del t, where x is between 0 and pi, t is greater than 0.

00:20 Initial, the boundary condition is u t 0 equals to 0, and del u over del x at the point pi comma t is 0.

00:32 Then we have u of x comma 0 is f of x, and x is in between 0 and pi.

00:40 We need to find out the steady state solution.

00:44 So we will find the steady state solution.

00:49 So let's get started with that.

00:54 So first of all, we will let u x be steady state solution.

01:01 The steady state solution.

01:07 So what we will get from here.

01:09 So u double dash plus alpha e raised to minus x equals to 0.

01:14 This is from equation number one.

01:17 Ok, so that is what we have been given here.

01:21 So u double dash comes out to be minus alpha e raised to minus x.

01:26 So from this, what i will get from this, if we integrate it.

01:35 So this will be u dash is equals to integral of minus alpha e raised to minus x dx.

01:46 So this is by integration.

01:50 Ok, we have a equation, we are getting this.

01:54 This will be u dash equals to minus alpha.

01:58 And again, we will get minus will become plus e raised to minus x plus here i will get some constant...

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