Curve-fitting of experimental data is one thing. Solving pde is another thing. What your problem exactly?

Dear Allia Anisa,

A problem very similar (zero Neumann boundary conditions) to your is completely solved at:

1. http://home.agh.edu.pl/~pgrab/grabowski_files/hypertest/coursessu22.xml#x42-290005.2

2. http://home.agh.edu.pl/~pgrab/grabowski_files/lecturedition2/newlecturese31.xml#x53-520007.5

Regards

dear shiuh-hwa, i want to do curve fitting. but the model is in pde form. so i would like to know if i need to solve pde first since i cannot make a function in matlab if the model in pde form. please correct me if i am wrong

dear grabowski, thank you for the links. 

I'll say do them separately. Solve pde and compare the results with the experimental data. For the experimental data, they can be fitted using least-square. In matlab, use polyfit to find the least-square fit function and use polyval to caculate using the function.

The general idea it’s to build a function that solves the pde, this must be a function of time, the space variable and the parmeters to be fitted f(t,x,par). If you have the analitic solution to the pde then building the function will be relatively easy, but if you are going to use a numerical method to solve the pde the construction will be more elaborated.

Once you have this function (by what i read in the matlab documentation),you can use directly the lsqcurvefit function to fit the data.

If the lsqcurvefit do not work you can fit the data using least squares were you minimize the function

(f(t_data,x_data,pars)-data)^2, essentially is the same approach of the lsqcurvefit, but you can use different optimization methods such as Levenberg-Marquard, Conjugated-Gradient (gradient driven methods), differential evolution, simulated annealing (global optimization methods).

I atteched you a file with some code that I used to solve a similar problem in wolfram mathematica,i hope you can find it useful.

dear octavia and shiuh-hwa, thank you for your answer. it helps me a lot.

Dear Illia,

I wonder that my links are not useful to you. Upon taking a=1, b=0, D=a, k=R_{a} your problem is exactly the same as in my example.

The characteristic ingredients of my approach are: introduce a step control into your second (nonzero) b.c. Arrange measurements of temperature at the opposite end (in control enginnering this is an identtification via the system step response). Find an almost exact solution using expansion w.r.t. ONB of eigenvectors of the open--loop system state operator (unforced system). Apply the least - squares method to compare the experimental data with theoretical one, computed from the expansion. This is an easy task as the control and observation are both of point nature. Be careful, the performance index of the least--squares may have several local minima, so this is a problem of global optimization.

Piotr

dear grabowski, thank you for the explanation. may i ask you something. actually before this i tried to solve pde of my model using finite differential method. but i have problem on solving the reaction part, ku as attached. or should i use finite element method instead of fdm? i am stuck on that part  

Dear Illia,

For the time being I can only say that you may eliminate the term $-ku$ by observing that if $t\longmapsto u$ is a solution of the full system then, $t\longmapsto e^{-kt}u(t)$ is a solution of the system with $k=0$.

Regards, Piotr

dear grabowski, the term of -ku is expressed the reaction during the process. i dont think assume it will be eliminated at prolonged time (full system). attached the journal that also used the same model. thanks for your quick response. 

dear Illia, can you talk more about your finite difference? there might be something in your code, like bug,scheme or something. Let's see how your simulation goes.

dear shyu, i tried to solve pde using finite differential method. for du/dt and d^2u/dr^2, i used central differential, but i dont know how to solve term of ku. if i am not mistaken, to solve the overall pde with boundary conditions, i need to use separation variables if the boundary in homogeneous. but i dont think that the boundary conditions of mine is homogeneous. the biggest problem is i am stuck on term of ku. example of du/dt i can solve as ui,j....please correct me if i am wrong.

You cannot use central difference for du/dt. Either backward or forward. Using central difference for du/dt is unstable with/without k*u term.

For the separation variable, if you choose to use finite difference, you don't use separation of variable.

dear shiuh-hwa, thank you for that information. as suggested by grabowski, i can use method of lines to solve the pde right?

regards

Dear Ilia,

There are too many methods to solve your problem. FEM, FD, spretral analysis, or even more exact approach, the eigen-function expansion. It's up to your choice. Just do it right.