Question

Solve the following partial differential equations. 1. Heat Equation 4 ?²u/?x² = ?u/?t 0 < x < ?, t > 0 Boundary condition u(0, t) = 0 t > 0 u(?, t) = 0 t > 0 Initial condition u(x, 0) = f(x) 0 ? x ? ? 2. Wave Equation 9u_xx = u_tt 0 < x < ? t > 0 u(0, t) = 0 t > 0 u(?, t) = 0 t > 0 u(x, 0) = f(x) 0 ? x ? ? where, f(x) = {x 0 ? x ? ?/2, ? - x ?/2 ? x ? ? u_t(x, 0) = 0 0 ? x ? ?

          Solve the following partial differential equations.

1. Heat Equation
4 ?²u/?x² = ?u/?t  0 < x < ?, t > 0
Boundary condition  u(0, t) = 0  t > 0
  u(?, t) = 0  t > 0
Initial condition  u(x, 0) = f(x) 0 ? x ? ?

2. Wave Equation
9u_xx = u_tt  0 < x < ?  t > 0
  u(0, t) = 0  t > 0
  u(?, t) = 0  t > 0
  u(x, 0) = f(x)  0 ? x ? ?
where,  f(x) = {x 0 ? x ? ?/2, ? - x ?/2 ? x ? ?
  u_t(x, 0) = 0  0 ? x ? ?

Solve the following partial differential equations.

1. Heat Equation
4 ?²u/?x² = ?u/?t 0 < x < ?, t > 0
Boundary condition u(0, t) = 0 t > 0
u(?, t) = 0 t > 0
Initial condition u(x, 0) = f(x) 0 ? x ? ?

2. Wave Equation
9uxx = utt 0 < x < ? t > 0
u(0, t) = 0 t > 0
u(?, t) = 0 t > 0
u(x, 0) = f(x) 0 ? x ? ?
where, f(x) = x 0 ? x ? ?/2, ? - x ?/2 ? x ? ?
ut(x, 0) = 0 0 ? x ? ?

Added by Charles M.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Madhur L

07/10/2023

Step 1

The heat equation is given by: $$u_t = a u_{xx}$$ with boundary conditions: $$u(0,t) = 0, \quad t > 0$$ $$u(T,t) = 0, \quad t > 0$$ and initial condition: $$u(x,0) = f(x), \quad 0 < x < T$$ We can solve this equation using separation of variables. Let Show more…

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