Need help to create this equation

Hi everyone, I am a little bit confused about this

Could someone helping me to create this? It seems that to many partial derivatives, since I am not good at it.

*you can assume the parameter by yourself or let them be in coefficient

Hope someone could help me to understand clearly how to perform this equation. Thankyou very much!

[Kevin Isakayoga]

can somebody help me?

Thanks

[Kevin Isakayoga]

@darova @Bjorn Gustavsson

Answer 1

[Walter Roberson]

syms rho_eff_hcp
syms p_w(x,t)
syms w(p_w_)
syms w_r(t)
dw_dp_w = diff(w(p_w_),p_w_)

dw_dp_w = 

dp_w_dt = diff(p_w(x,t),t);
LHS = 1/rho_eff_hcp * dw_dp_w * dp_w_dt;
dp_w_dx = diff(p_w(x,t),x)

dp_w_dx = 

syms K
RHS_temp1 = K*w(p_w_)*dp_w_dx

RHS_temp1 = 

RHS_temp2 = diff(RHS_temp1,x)

RHS_temp2 = 

dw_r_dt = diff(w_r(t),t);
RHS = -RHS_temp2 + dw_r_dt;

eqn = LHS == RHS

eqn = 

Now you are faced with the problem that w should be a function of p_w instead of p_w_ but p_w is a function. You are trying to take the derivative of a function with respect to a function, which requires Calculus of Variations instead of regular calculas.

You also have three functions but only one equation. There is no hope that you would be able to resolve this symbolically, even if the equations were ODE instead of PDE.

[Kevin Isakayoga]

            RHH(I,J+1)=(((r*(timeh(J+1)-timeh(J))/(24))*(rhoeff)*(1/dwrht(J))*p0*(Kw(J+1)*(RHH(I+1,J+1)-2*RHH(I,J+1)+RHH(I-1,J+1))+Kw(J)*(RHH(I+1,J)-2*RHH(I,J)+RHH(I-1,J))+0))+RHH(I,J));
 
 

maybe that is my thinking for now, is it correct if I use the crank nicolson method like this?

[Walter Roberson]

Sorry, I am not familiar with Crank Nicolson.

I should probably modify the above code a bit:

syms rho_eff_hcp
syms p_w(x,t)
syms w(p_w_,x,t)
syms w_r(t)
dw_dp_w = diff(w(p_w_,x,t),p_w_)

dw_dp_w = 

dp_w_dt = diff(p_w(x,t),t);
LHS = 1/rho_eff_hcp * dw_dp_w * dp_w_dt;
dp_w_dx = diff(p_w(x,t),x)

dp_w_dx = 

syms K
RHS_temp1 = K*w(p_w_,x,t)*dp_w_dx

RHS_temp1 = 

RHS_temp2 = diff(RHS_temp1,x)

RHS_temp2 = 

dw_r_dt = diff(w_r(t),t);
RHS = -RHS_temp2 + dw_r_dt;

eqn = LHS == RHS

eqn = 

[Kevin Isakayoga]

Thank you for your response! However i'm trying to solve it with crank nicolson method

[Bjorn Gustavsson]

The key point here is, as Walter mentions, that you seem to have 3 functions, , w and but only one equation. That most likely will have an infinite number of solutions (only a guess, but based on the idea that you might chose any function for one of the three and most likely can chose a reasonable number of other functions for one of the remaining two and still get a solution for the third). So unless you have some explicit information given about two of the functions or their relation with the third there doesn't seem to be enough to get a unique solution or even a set of solutions.

[Kevin Isakayoga]

clear; clc;
days=36500;
var(1)=0.001;
timeinit=0.001;
counter=2;
while (timeinit<days)
    timeinit=timeinit*1.1;
    var(counter)=timeinit;
    counter=counter+1;
end
timeh=zeros(1,length(var)+1);
timeh1=zeros(1,length(var)+2);
timeh(1)=0; %should be changed
for m=1:length(var)
timeh(m+1)=var(m)*24;
timeh1(m+1)=var(m)*24;
end
timeday=timeh/24;
RH1=linspace(0.005,1,length(timeday));
vmbet=2.72*10^-4;
Cbet=18;
kbet=0.83;
casi=1.8;                    %atomic ratio
%sh20=202.6;                 %m2/g
wc=0.5;         %water to cement ratio
T=20;           %celcius
Tcal=T+273.15;
rhoeff=2.04;                 %density of HCP at 105C g/cm3
sh20=234-297*log(casi);      %m2/g
Sad=sh20*rhoeff*exp(-0.017*(Tcal-293));

cm=10;                    % Prediction length
h=0.1;                    % the interval of distance (cm)
M=(cm)*(1/h);             % Cut-off line of x-axis (days)
E=0.00001;
for mm=1:length(RH1)
    if (RH1(mm)<=0.98)
        dwrht(mm)=(Cbet*RH1(mm)*Sad*kbet^2*vmbet)/((RH1(mm)*kbet*(Cbet - 1) + 1)*(RH1(mm)*kbet - 1)^2) - (Cbet*Sad*kbet*vmbet)/((RH1(mm)*kbet*(Cbet - 1) + 1)*(RH1(mm)*kbet - 1)) + (Cbet*RH1(mm)*Sad*kbet^2*vmbet*(Cbet - 1))/((RH1(mm)*kbet*(Cbet - 1) + 1)^2*(RH1(mm)*kbet - 1));
    else
        dwrht(mm)=(Cbet*RH1(mm)*Sad*kbet^2*vmbet)/((RH1(mm)*kbet*(Cbet - 1) + 1)*(RH1(mm)*kbet - 1)^2) - (Cbet*Sad*kbet*vmbet)/((RH1(mm)*kbet*(Cbet - 1) + 1)*(RH1(mm)*kbet - 1)) + (Cbet*RH1(mm)*Sad*kbet^2*vmbet*(Cbet - 1))/((RH1(mm)*kbet*(Cbet - 1) + 1)^2*(RH1(mm)*kbet - 1));%50*w0293 - 50*w98ad;
    end
end
ah=1.05-3.8*wc+3.56*(wc)^2;
bh=-14.4+50.4*wc-41.8*(wc)^2;
gamh=31.3-136*wc+162*(wc)^2;
Dh=ah+bh.*(1-2.^(-10.^(gamh*(RH1-1))));
RH=0.766; %relative humadity
RHH=zeros(M+1,length(var));
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% boundary condition %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for J=1:length(var)+1
    RHH(:,1)=1;
    if (J==1)
        RHH(1,J)=RH;
    else
        RHH(1,J)=RH;
    end
    RHH(M+1,J)=RH;                %it's a boundary condition
end
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Crank¡VNicolson method %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 r=1/(2*h^2);
for J=1:length(var)
    for I=2:M
        RHH(I,J+1)=RHH(I,J);
    end
    for L=1:200
        LL=L;                   %number of times of calculation
        for I=2:M
            UF(I,J+1)=RHH(I,J+1);
            RHH(I,J+1)=(((r*(timeh(J+1)-timeh(J))/(24))*(1/dwrht(J))*Dh(J)*((RHH(I+1,J+1)-2*RHH(I,J+1)+RHH(I-1,J+1))+(RHH(I+1,J)-2*RHH(I,J)+RHH(I-1,J))+0))+RHH(I,J));
        end
        S=0;
        SO=0;
        for I=2:M
            SO=SO+abs(RHH(I,J+1));
            S=S+abs(RHH(I,J+1)-UF(I,J+1));
        end
        S=S/SO;
        if(S<=E)
            break;
        end
    end
end

maybe this is the code, but my equation didnt show a good results based on the following equation

and I want to get results like I mentioned below

and this is the boundary condition

[Bjorn Gustavsson]

Don't programme like this. Separate this into a setup-running-script and a C-N function.

Write a function for the C-N integration of your PDE that should take the necessary initial and boundary functions, the source-functions and coefficients as input arguments. In that function you create the required LHS and RHS matrices and whatnot. That way you can verify that it works and you get a clean smaller function to use. This is many many lines of initialization and generation of everything and then a, seemlingly, hard-coded integration-step. I cannot read this and see whether it is a sensible solution, sorry too many lines.

[Kevin Isakayoga]

I have fixed it with a brief code

Hope you could take a look, thank you for your response.

[Kevin Isakayoga]

I'm pretty sure that all of my coefficient such as dh/dw and Dh are correct.However, I just confuse my calculation and maybe my boundary condition was wrong.

[Bjorn Gustavsson]

No, it doesn't look right. When you write your C-N function you will get to a stage where you have to multiply the RHS with the inverse of the LHS-matrix. You have neither an inv nor a \ in your C-N section. Take a pen and paper, sit down and write out the expressions for the LSH and RHS matrices and then implement those expressions. You will get a step looking something lke this:

I_next = invMLHS*([Mrhs]*I_prev + Q_t_p_half);

HTH