Transform the following IBVP into a problem with homogeneous boundary conditions.
$$ \begin{array}{l} u_{t t}=u_{x x}, \quad 0<x<1, t=>0 \\ a u(0, t)+b u_{x}(0, t)=f_{1}(t) \quad \text { and } \quad c u(1, t)+d u_{x}(1, t)=f_{2}(t), \quad t>0 \\ u(x, 0)=g(x), \quad u_{t}(x, 0)=h(x), \quad 0<x<1 \end{array} $$
where \(a, b, c\) and \(d\) are constants.
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Transform the following IBVP into a problem with homogeneous boundary conditions.
(35 marks) The vibration of a semi-infinite string is described by the following initial boundary value problem.(35 marks) The vibration of a semi-infinite string is described by the following initial boundary value problem.$$ \begin{array}{l} u_{t t}=c^{2} u_{x x}, \quad 0< x < \infty, t>0 \\ u(x, 0)=A e^{-\alpha x} \quad \text { and } \quad u_{t}(x, 0)=0, \quad 0< x < \infty \\ u(0, t)=A \cos \omega t, \quad t>0 \\ \lim _{x \rightarrow \infty} u(x, t)=0, \quad \lim _{x...
(25 marks) Solve the following initial value problem using Fourier transform.$$ \begin{array}{l} u_{t}=u_{x x}, \quad-\infty< x <\infty, t= >0 \\ u(x, 0)=\left(1-2 x^{2}\right) e^{-4 x^{2}}, \quad-\infty< x <\infty \end{array} $$with \(u(x, t) \rightarrow 0\) and \(u_{x}(x, t) \rightarrow 0\) as \(x \rightarrow \pm \infty\).
Solve the following IBVP by eigenfunction expansion.$$ u_{t t}=u_{x x}+1+t \cos (\pi x), \quad 0<x<1, quad="" t="">0 $$$$ u_{x}(0, t)=0 \quad \text { and } \quad u_{x}(1, t)=0, \quad t>0 $$$$ u(x, 0)=2 \quad \text { and } \quad u_{t}(x, 0)=-2 \cos (2 \pi x), \quad 0<x<1 $$
Solve the following initial boundary value problem using Laplace transform.$$ \begin{aligned} u_{t} &=u_{x x}+t e^{-\pi^{2} t} \sin (\pi x), & 0<x<1, t="">0 \\ u(0, t)=0, & u(1, t)=0, & t>0 & \\ u(x, 0) &=\sin (2 \pi x) & & \end{aligned} $$
nonhomogeneous vibrating string problem for u(x with homogeneous boundary conditions t > 0 u(0, t) u(r,t) = 0, 0, = and the initial conditions 0stst tr(z,0)=0, u(z, 0) sin(2x), = Find the solution u(x,t) to the IBVP using an eigenfunction expansion: u(z, t) = Σ an(t) sin(nz) n-1 nonhomogeneous vibrating string problem for u(x with homogeneous boundary conditions t > 0 u(0, t) u(r,t) = 0, 0, = and the initial conditions 0stst tr(z,0)=0, u(z, 0) sin(2x), = Find the...
5. Consider the following IBVP (initial boundary value problem utt - Curr = 0, 0<x<1, t>0, with boundary conditions u(0,t) = u(1, t) = 0, > 0 and initial conditions (7,0) = x(1 – 2), 14(2,0) = 0, 0<x< 1. Use separation of variables method to find an infinite series solution of this problem. Do a complete calculation for this problem.
(1 point) Solve the heat problem with non-homogeneous boundary conditions ди (x, t) at = a2u (2,t), 0 < x < 5, t> 0 ar2 u(0,t) = 0, u5,t) = 3, t>0, u(x,0) = **, 0<x< 5. Recall that we find h(x), set v(x, t) = u(x, t) – h(x), solve a heat problem for v(x, t) and write u(x, t) = v(x, t) +h(x). Find h(c) h(x) = The solution u(x, t) can be written as u(x,t) =h(x) +...
2. In lectures we solved the heat PDE in 1 +1 dimensions with constant-temperature boundary conditions u(0,t)u(L,t) -0. If these boundary conditions change from zero temperature, we need to do a little bit more work. Consider the following initial/boundary-value problem (IBVP) 2 (PDE) (BCs) (IC) u(0,t) = a, u(x,00, u(L, t)=b, st. and let's take L = 1, a = 1, b = 2 throughout for simplicity. Solve this problem using the following tricks b and A"(x)-0 (a) Find a...
(1 point) Solve the heat problem with non-homogeneous boundary conditions du (x, 1) = ot (x,1), 0<x<2, t> 0 dx (0,t) = 0, (2, 1) = 2, t> 0, u(x,0) = 0<x<2. Recall that we find h(x), set u(x, t) = u(x, t)-h(x), solve a heat problem for u(x, t) and write u(x, t) = u(x, t) + h(x). Find h(x) h(x) = The solution u(x, t) can be written as u(x, t) = h(x) + u(x, t), where u(x,...
3. This question is about non-homogeneous boundary conditions (a) Consider Laplace's equation on a rectangle, with fully inhomogeneous boundary conditions =0 0 a, 0< y <b u(x, 0) fi() u(, b) f2(a) u(0, y)g (x) ua, y) = 92(r) 0 ra Homogenise the boundary conditions to convert the problem to one of the form 2 F(x, y) 0 xa,0 y < b + (x, 0)= fi() b(x, b) f2(x) b(0, y)0 (a, y) = 0 0y b 0 y sb...