Teaching CSIR NET Mock Test Series Mathematical Science Partial Differential Equations Laplace Equation, Heat and Wave Equations
Let u(x, t) be a function that satisfies the PDE
Uxx - Utt = ex + 2t, x ∈ ℝ, t > 0 and the initial conditions
u(x, 0) = cos(x), Ut(x, 0) = 0 for every x ∈ ℝ. Here subscripts denote partial derivatives corresponding to the variables indicated. Then the value of u \(\left(\pi/2, \pi/2 \right)\) is
1
\(\rm e^{\pi / 2}\left(1+\frac{1}{2} e^{\pi / 2}\right)+\left(\frac{\pi^3+4}{8}\right)\)
2
\(\rm e^{\pi / 2}\left(1+\frac{1}{2} e^{\pi / 2}\right)+\left(\frac{\pi^3-4}{8}\right)\)
3
π /4 - eπ /2
4
\(\rm e^{\pi / 2}\left(1-\frac{1}{2} e^{\pi / 2}\right)-\left(\frac{\pi^3-4}{8}\right) \)