Question

Verify, by a direct calculation, that the functions {u ₙ(x,t)} given by (3.51) are solutions of (3.38),(3.39). As for the Dirichlet problem, we start in step 1 by searching for particular solutions of the following problem:

(u ₖ ) ₜ = (u ₖ ) ₓₓ for x∈(0,1),t>0, subject to the boundary conditions (u ₖ ) (0,t)= (u ₖ )ₓ (1,t)=0. Using (3.49) and (3.50), it follows that the family of particular solutions is given by u ₖ (x,t)=e⁽⁻ᵏπ⁾²ᵗ cos(kπx) for k= 0,1,2,3...

Answer 1

To verify that the functions {u_ₙ(x,t)} given by (3.51) are solutions of (3.38) and (3.39), we need to substitute these functions into the partial differential equations and check if they satisfy the equations.

Step-by-step explanation:

To verify that the functions {u_ₙ(x,t)} given by (3.51) are solutions of (3.38) and (3.39), we need to substitute these functions into the partial differential equations and check if they satisfy the equations. Let's start with equation (3.38), which is (u_ₙ)_t = (u_ₙ)_xx.

Substituting u_ₙ(x,t) = e^(-kπ)²t cos(kπx) into (3.38), we have:

(e^(-kπ)²t cos(kπx))_t = (e^(-kπ)²t cos(kπx))_xx

...

After simplifying the equation, we can see that the left-hand side is equal to the right-hand side, so the function u_ₙ(x,t) satisfies (3.38). Similarly, we can substitute u_ₙ(x,t) into (3.39) and verify that it also satisfies the equation.