How to solve this equation for Laplace transform with matlab?

[darova]

What do you mean by 'solving'? Is it a surface? What are those equations?

[soe min aung]

This equation is the sea floor displacement function. I want to solve this equation to obtain the free surface elevation. So, we need to transform this equation by using Fourier-Laplace transform with matlab. Please....

[darova]

Did you try to build those surface/equations? Can you show your attempts?

Where is the problem?

[soe min aung]

clear
clc
syms eta0 L v t x y k k1 k2 s h w
% eta0 = 2 ; 
% L = 100 ; 
% W = 100 ; 
% v = 0.14 ; 
% h = 2;
% g = 0.0098;
% t = 5.95;
T1 = (eta0*v*t)/(2*L)*(1-cos(pi/50*x)).*(1-cos(pi/100*(y+150))) ; 
T2 = (eta0*v*t)/L*(1-cos(pi/50*x)) ;
T3 = (eta0*v*t)/(2*L)*(1-cos(pi/50*x)).*(1-cos(pi/100*(y-150))) ; 
T=T1+T2+T3;
%Laplace transform of 'T'.
Lap = laplace(T)
%Double Fourier transform of 'Lap' for 'x' and 'y'.
eta_1 = fourier(Lap,x,k1)
eta_ = fourier(eta_1,y,k2)
%Substituting 'eta_' into 'zeta_1' formula.

% w = sqrt(g*k*tanh(k*h))

zeta_1 = (eta_.*s^2)/(cosh(k*h)*(s^2+w^2))
%Inverse Laplace transform of 'zeta_1'.
zeta_ = ilaplace(zeta_1,s,t)
%Double inverse Fourier transform of 'zeta_' for 'k1' and 'k2'.
zeta1 = ifourier(zeta_,k1,x)
zeta = ifourier(zeta1,k2,y)
%For figure.
x=0;100;
y=linspace(-200,200,101)
[X,Y] = meshgrid(x,y) ;
surf(X,Y,zeta)

I can't figure.

[Walter Roberson]

Since you are using the symbolic toolbox anyhow, use piecewise() to define your equation. Then when you have it in piecewise form, ude rewrite() specifying the 'Heaviside' option. With the equation in heaviside form, the symbolic toolbox laplace transform functions can create appropriate expressions.

[soe min aung]

Could you please help me an example code for this problem? Please sir....

[Walter Roberson]

syms eta0 L v t x y k k1 k2 s h w

T1 = (eta0*v*t)/(2*L)*(1-cos(pi/50*x)).*(1-cos(pi/100*(y+150))) ; 
T2 = (eta0*v*t)/L*(1-cos(pi/50*x)) ;
T3 = (eta0*v*t)/(2*L)*(1-cos(pi/50*x)).*(1-cos(pi/100*(y-150))) ;
R = @(V,L,H,X)heaviside(V-L)*heaviside(H-V)*X

T = simplify( R(x,0,100,R(y,-150,-50,T1) + R(y,-50,50,T2) + R(y,50,150,T3)) )

Now you can do laplace() on T. However, when you do so, you need to specify which variable you are transforming with respect to. Likewise when you fourier, make sure you do so with respect to the correct variable.

Also, you need to carefully examine what happens at the exact boundaries.

[soe min aung]

Thanks you so much for your kindness. When I was calculating your advice, I got a result but not figure. I want to get 3D figure. But I cannot try it. Please help me sir. How to plot for 3D?

syms eta0 L v t x y k k1 k2 s h w
eta0 = 2 ; 
L = 100 ; 
W = 100 ; 
v = 0.14 ; 
h = 2;
g = 0.0098;
t = 5.95;
T1 = (eta0*v*t)/(2*L)*(1-cos(pi/50*x)).*(1-cos(pi/100*(y+150))) ; 
T2 = (eta0*v*t)/L*(1-cos(pi/50*x)) ;
T3 = (eta0*v*t)/(2*L)*(1-cos(pi/50*x)).*(1-cos(pi/100*(y-150))) ;
R = @(V,L,H,X)heaviside(V-L)*heaviside(H-V)*X
T = simplify( R(x,0,100,R(y,-150,-50,T1) + R(y,-50,50,T2) + R(y,50,150,T3)) )
Lap = laplace(T)
%Double Fourier transform of 'Lap' for 'x' and 'y'.
eta_1 = fourier(Lap,x,k1)
eta_ = fourier(eta_1,y,k2)
%Substituting 'eta_' into 'zeta_1' formula.

w = sqrt(g*k*tanh(k*h))

zeta_1 = (eta_.*s^2)/(cosh(k*h)*(s^2+w^2))
%Inverse Laplace transform of 'zeta_1'.
zeta_ = ilaplace(zeta_1,s,t)
%Double inverse Fourier transform of 'zeta_' for 'k1' and 'k2'.
zeta1 = ifourier(zeta_,k1,x)
zeta = ifourier(zeta1,k2,y)
%For figure.
x=0;100;
y=linspace(-200,200,101)
[X,Y] = meshgrid(x,y);
surf(X,Y,zeta)

[Walter Roberson]

Your zeta is a symbolic expression in k and y according to symvar(), but it also contains k2 inside fourier() calls.

Perhaps you should do

kvals = 0:100;
yvals = linspace(-200,200,101);
[K,Y] = meshgrid(kvals, yvals);
Zsym = subs(zeta, {k, y}, {K, Y});
Z = double(Zsym);
surf(K,Y,Z)

However in practice this does not work: after the substitution, double() is not able to resolve some of the values to numeric. I discovered that some of them can be converted to a better form by using simplify(Zsym), but that is slow.

[soe min aung]

Thank you so much sir, I want to try it. If I did not know, let me ask you.