Complex Variables and Partial Differential Equations (3130005)

Question-5c-OR

BE | Semester-3 Summer-2020 | 27-10-2020

Q5) (c)

Find the solution of $u_{t} = c^{2} u_{tt}$ together with the initial and boundary conditions $u (0, t) = u (2, t) = 0; t \geq 0, u (x, 0) = 10; 0 \leq x \leq 2 .$

Solution of Heat equation:

u (x, t) = \sum_{n = 1}^{\infty} B_{n} \sin \frac{nπx}{L} e^{- \frac{n^{2} π^{2} c^{2} t}{L^{2}}}

B_{n} = \frac{2}{L} \int_{0}^{L} f (x) \sin \frac{nπx}{L} dx

B_{n} = \frac{2}{2} \int_{0}^{2} 10 \sin \frac{nπx}{2} dx

B_{n} = 10 {[\frac{- \cos \frac{nπx}{2}}{\frac{nπ}{2}}]}_{0}^{2}

B_{n} = \frac{20}{nπ} [- \cos nπ + \cos 0] = \frac{20}{nπ} [1 - \cos nπ]

B_{n} = \frac{20}{nπ} [1 - {(- 1)}^{n}]

Solution:

u (x, t) = \sum_{n = 1}^{\infty} B_{n} \sin \frac{nπx}{L} e^{- \frac{n^{2} π^{2} c^{2} t}{L^{2}}}

u (x, t) = \sum_{n = 1}^{\infty} \frac{20}{nπ} [1 - {(- 1)}^{n}] \sin \frac{nπx}{2} e^{- \frac{n^{2} π^{2} c^{2} t}{4}}

Questions

Go to Question Paper