Hi Matterazzi,
I understand that you would like to solve the following Fredholm equation of the first kind for f(t)
You can use numerical methods such as Tikhonov regularization or the inverse Laplace transform.
- Tikhonov regularization is a common approach to solve ill-posed inverse problems. It involves introducing a regularization parameter to stabilize the solution. In MATLAB, you can use the “lsqnonneg” function to solve the regularization problem. This function performs non-negative least squares optimization, which is suitable for non-negative solutions
- If you have access to the Laplace transform of the function F(t), you can use the inverse Laplace transform to obtain the solution f(t). In MATLAB, you can use the “ilaplace” function to perform the inverse Laplace transform.
Here's an overview of how you can approach this problem using MATLAB:
- Define the problem: Start by defining the function F(t) and the integration domain D. Determine the values of time (t) for which you have data for F(t).
- Discretize the integral equation: Approximate the integral equation using a numerical method. For example, you can use numerical integration techniques such as the Trapezoidal rule or Simpson's rule to discretize the integral.
- Formulate the linear system: Convert the integral equation into a linear system of equations. This can be done by evaluating the discretized integral equation at different time points. The unknown function f(k) can be represented as a vector of values at different k points.
- Solve the linear system: Once you have a system of linear equations, you can apply Tikhonov regularization which involves adding a regularization term to the system. You can then use the “lsqnonneg” function in MATLAB to solve the regularized least squares problem. Alternaively, if you’d like you can use “ilaplace” function for Inverse Laplace transform approach.
To know more you can refer to the documentation links attached below:
