7.3 Wave Equation 413
- Compare the results of Exercise 1 with the d’Alembert solution.
- Obtain an approximate solution of Eqs. (1), (2), and (3) withf(x)≡0and
g(x)=sin(πx).Take x= 1 /4,ρ=1. - Compare the results of Exercise 3 with the exact solutionu(x,t)=
( 1 /π )sin(πx)sin(πt). - Obtain an approximate solution of Eqs. (1), (2), and (3) withg(x)≡0and
f(x)asinEq.(8).Use x= 1 /4andρ^2 = 1 /2. - Compare the entries of Table 6 with the d’Alembert solution.
- Obtain an approximate solution of this problem with a time-varying
boundary condition, using x= t= 1 /4.
∂^2 u
∂x^2 =
∂^2 u
∂t^2 ,^0 <x<^1 ,^0 <t,
u( 0 ,t)= 0 , u( 1 ,t)=h(t), 0 <t,u(x, 0 )= 0 ,∂u
∂t(x,^0 )=^0 ,^0 <x<^1 ,h(t)={ 1 , 0 <t<1,
− 1 , 1 <t< 2
andh(t+ 2 )=h(t),h( 0 )=h( 1 )=0.- Same task as Exercise 7 but√ h(t)=sin(πt). Use sin(π/ 4 )∼= 0 .7 instead of
2 /2. - Find starting and running equations for the following problem. Using
x= 1 /4, find the longest stable time step and compute values of the
approximate solution formup to 8.
∂^2 u
∂x^2 =
∂^2 u
∂t^2 +^16 u,^0 <x<^1 ,^0 <t,
u( 0 ,t)= 0 , u( 1 ,t)= 0 , 0 <t,u(x, 0 )=f(x), ∂u
∂t(x, 0 )= 0 , 0 <x< 1 ,wheref(x)is given in Eq. (8).10.Using x= 1 /4andρ^2 = 1 /2, compare the numerical solution of the
problem in Exercise 9 with and without the 16uterm in the partial differ-
ential equation.