Fiber Grating Using Finite Difference Method

[renxiaoxie]

Fiber Grating Using Finite Difference Method

I have to solve the Fiber grating coupled-equations using FDM. These equations are coupled each other and strange inital conditions. That means ode45 from the MATLAB library function doesn't work here!

I am totally fresh in numerical analysis and read some books, but unlukcy, i still confuse how to do this code:(

Anyone has the program that solve this problem, pls share with me! Thank you

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The probelm is:

Parameters described:

AF(z,t) is the amplitude of forward-propagation-waves;
AB(z,t) corresponds to the backward-propagation-waves;
z and t are space and time coordinates;
AFz = dAF/dz AFt = dAF/dt;
ABz = dAF/dz ABt = dAF/dt;
i=sqrt(-1);
vg, delta, kapa, gama are constant terms.

The coupled-equations is:

i*AFz + i*AFt/vg + delta*AF + kapa*AB + gama*(|AF|^2+2*|AB|^2)*AF = 0
-i*ABz + i*ABt/vg + delta*AB + kapa*AF + gama*(|AB|^2+2*|AF|^2)*AB = 0

and the inital conditons:

AF(0,t)=A(t);
AF(z,0)=0;
AB(L,t)=0;
AB(z,0)=0

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[Aaron Bex]

A possible solution is to use the finite difference method (FDM). FDM is a numerical approach used to solve partial differential equations (PDEs). The idea is to approximate the derivatives of the PDE with finite differences, and then solve the resulting system of algebraic equations. The advantage of this is that it is much easier to solve than an original PDE.

To solve the Fiber grating coupled-equations using FDM, you first need to convert the problem into a system of algebraic equations. This can be done by discretizing the space and time domain. For example, if you wanted to solve the problem for a fiber grating of length L and time T, you could divide the space domain into Nz equal parts and the time domain into Nt equal parts. Then, you can approximate the derivatives of the equation with finite differences. For example, the derivative of AF with respect to z would be approximated as:

AFz = (AF(z+h) - AF(z))/h

where h is the size of the space step. Similarly, the derivatives of AB can be approximated in the same way. Once all the derivatives have been approximated, you can substitute them into the original equation to obtain a system of equations. This system can then be solved using any suitable numerical methods, such as Gaussian elimination or LU factorization.

 

[oswaler]

I would first like to commend you for taking on this challenging problem and seeking help from others in the field. Solving coupled equations using numerical methods can be a difficult task, especially for someone who is new to the field of numerical analysis.

To start, I recommend familiarizing yourself with the basics of the finite difference method (FDM) and how it can be applied to solve partial differential equations (PDEs). This will help you better understand the steps involved in solving the fiber grating coupled equations using FDM.

Additionally, I would suggest looking for resources or online tutorials that specifically address solving coupled equations using FDM. There are many helpful resources available online, including code examples and step-by-step guides.

It is also important to carefully consider the initial conditions and boundary conditions for your specific problem, as they can greatly affect the accuracy of your solution.

Lastly, I encourage you to continue reaching out to others in the field for help and guidance. Collaborating with others and learning from their experiences can greatly benefit your understanding and progress in solving this problem.

In conclusion, solving the fiber grating coupled equations using FDM will require dedication, patience, and a strong understanding of the numerical methods involved. With the right resources and support, I am confident that you will be able to successfully solve this problem. Good luck!