Question

what is the fomula of the numerical solution of the heat eqation

Ut = a² Uxx ? (not the analytical solution the numerical solution)

Give example please

Answer #1

Simplest numerical formula for 1D heat equation is:

U_{i}^{new =} U_{i}^{old} +
alpha*(U_{i+1}^{old} - 2U_{i}^{old}
+ U_{i-1}^{old})

where alpha=a*dt/dx^{2}

------ Here is a MATLAB code to generate the solution -------

% Solution of the Heat Equation Using a Forward Difference
Scheme

% Initialize Data

% Length of Rod, Time Interval

% Number of Points in Space, Number of Time Steps

L=1;

T=0.1;

a=1;

N=10;

M=50;

dx=L/N;

dt=T/M;

alpha=a*dt/dx^2;

% Position

for i=1:N+1

x(i)=(i-1)*dx;

end

% Initial Condition

for i=1:N+1

u0(i)=sin(pi*x(i));

end

% Partial Difference Equation (Numerical Scheme)

for j=1:M

for i=2:N

u1(i)=u0(i)+alpha*(u0(i+1)-2*u0(i)+u0(i-1));

end

u1(1)=0;

u1(N+1)=0;

u0=u1;

end

% Plot solution

plot(x, u1);

Solve the heat equation and find the steady state solution:
uxx = ut, 0 < x < 1, t > 0,
u(0,t) = T1, u(1,t) = T2, where T1
and T2 are distinct
constants, and u(x,0) = 0

Find the solution formula for the heat equation ut =
c2 uxx on the half-infinite bar (0 ≤ x <
∞) with Dirichlet boundary condition u(0, t) = a, for some constant
a, and initial condition u(x, 0) = f(x) using the Fourier sine
transform.

Solve the heat equation and find the steady state solution :
uxx=ut 0<x<1, t>0,
u(0,t)=T1, u(1,t)=T2, where T1 and T2 are
distinct constants, and u(x,0)=0

Use the Fourier sine transform to derive the solution formula
for the heat equation ut = c2 uxx
on the half-infinite bar (0 ≤ x < ∞) with Dirichlet boundary
condition u(0, t) = a, for some constant a, and initial condition
u(x, 0) = f(x).

Solve the heat equation ut = k uxx, 0 < x < L, t >
0
u(0, t) = u(L, t) = 0, t > 0
u(x, 0) = f(x), 0 < x < L
a) f(x) = 6 sin 9πx L
b) f(x) = 1 if 0 < x ≤ L/2 2 if L/2 < x < L

Solve ut=uxx, 0 < x < 3, given the
following initial and boundary conditions:
- u(0,t) = u(3,t) = 1
- u(x,0) = 0
Please write clearly and explain your reasoning.

Consider the one dimensional heat equation with homogeneous
Dirichlet conditions and initial condition:
PDE : ut = k uxx, BC : u(0, t) = u(L, t) = 0, IC : u(x, 0) =
f(x)
a) Suppose k = 0.2, L = 1, and f(x) = 180x(1−x) 2 . Using the
first 10 terms in the series, plot the solution surface and enough
time snapshots to display the dynamics of the solution.
b) What happens to the solution as t →...

What is the difference between an internal check and an external
check for a numerical solution? Give an example of an external
check.

uxx = ut - u (0<x<1, t>0),
boundary conditions: u(1,t)=cost, u(0,t)= 0
initial conditions: u(x,0)= x
i) solve this problem by using the method of separation of
variables. (Please, share the solution step by step)
ii) graphically present two terms(binomial) solutions for
u(x,1).

Determine the solution of the following initial boundary-value
problem
Uxx=4Utt 0<x<Pi t>0
U(x,0)=sinx 0<=x<Pi
Ut(x,0)=x 0<=x<Pi
U(0,t)=0 t>=0
U(pi,t)=0 t>=0

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