How to solve the following integral-differential equations?

[jamesbom100]

hello, everyone. Recently I confront with the integral differential equations(ide's) in my research. Anyone can help to solve h(t,s), A1(t,s), A2(t,s), B1(t,s), B2(t,s)?

The file attached as following. Thanks a lot. Numerical method is also okay!

 

[Mute]

Some of your equations look like they will be simplified if you Laplace transform them in s or t (or both).

The key rule is the convolution rule:

$$\mathcal L \left[ \int_0^t dt'~K(t-t')f(t')\right] = \hat{K}(s)\hat{f}(s).$$

Since some of your integrals are of this convolution form, Laplace transforming your equations looks like it may leave you with an algebraic equation for the Laplace transform of the function(s) you want to solve for.

 

[jamesbom100]

Hi, Dear Mute:
Thanks for your the key rule: Laplace transform of the convolution form. However, my beta function only has numerical values(obtained by matlab), not a closed form(because the upper and lower limit of the integration). So how can I get a Laplace transform for numerical values of the beta function(by any command in other package)? Otherwise, how can I do the inverse Laplace transform of this complex algebraic equation(because the Laplace transform of beta is very complex, I think), by the command residue in mathematica or others?

 

[Mute]

Hi, Dear Mute:
Thanks for your the key rule: Laplace transform of the convolution form. However, my beta function only has numerical values(obtained by matlab), not a closed form(because the upper and lower limit of the integration). So how can I get a Laplace transform for numerical values of the beta function(by any command in other package)? Otherwise, how can I do the inverse Laplace transform of this complex algebraic equation(because the Laplace transform of beta is very complex, I think), by the command residue in mathematica or others?

As the Laplace transform is just $\mathcal L [f(t)](y) = \int_0^\infty dt~\exp(-yt)f(t)$, you could easily solve for the Laplace transform numerically. The tricky part would probably be inverse transforming numerically.

Alternatively, you could fit a functional form to your $\beta$ function in terms of standard functions and compute the Laplace transform of the fit if you want to try and solve the equations analytically.

I suppose you could also combine both approaches and calculate the Laplace transform numerically and then fit a functional form to the Laplace transform of your $\beta$.

 

[blue_raver22]

Hello,

Thank you for reaching out for help with solving integral-differential equations (IDEs). These types of equations are commonly used in scientific research, and there are a few methods that can be used to solve them.

One approach is to use numerical methods, such as the Euler or Runge-Kutta methods. These methods involve approximating the solution to the IDE by dividing the problem into smaller steps and using iterative calculations to find the solution at each step. This method can be time-consuming and may not always provide accurate results, but it can be a good starting point for solving IDEs.

Another approach is to use analytical methods, such as separation of variables or Laplace transforms. These methods involve manipulating the equation algebraically to isolate the variables and then solving for them. This method can be more complex, but it can provide more accurate and precise solutions.

It is also important to note that the specific method used to solve an IDE may depend on the form of the equation and the boundary conditions given. Therefore, it is important to carefully examine the equation and any given information before deciding on a method to use.

I hope this helps you in solving your IDEs. If you need further assistance, I recommend consulting with a mathematics or physics expert who has experience with solving these types of equations.

Best of luck with your research!

Sincerely,