Integro-Differential Equation with mathematica

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In summary, the conversation discusses solving a system of Integro-Differential Equations in Mathematica, with a specific example provided and a suggestion to use the Euler method to compute the developing integral at each time step. There are no built-in commands for solving IDEs in Mathematica, but there may be numerical methods available in the Mathematica community.
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yashar
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hi
how can i solve a system of Integro-Differential Equations in mathematica numerically or analytically?
thanks
 
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yashar said:
hi
how can i solve a system of Integro-Differential Equations in mathematica numerically or analytically?
thanks

By starting off with just one. Say:

[tex]y'(x)=2-1/4 x^2+1/4\int_0^x y(t)dt,\quad y(0)=0[/tex]

Now, modify for example the Euler method so that at evey time step, compute the developing integral for example at time step [itex]x_k[/itex], compute (numerically)

[tex]\int_0^{x_k} y(t)dt[/tex]

where the values of y(t) are obtained from the previous calculations. Then just add that part to the regular calculations for that method. Try to write a Mathematica program to implement this and see if you come out with [itex]y(x)=2x[/itex]. Get that perfected, then move on to more complicated ones. There are no built-in commands to compute IDEs in Mathematica although I'm sure the Mathematica community has written some numerical ones.
 
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1. What is an integro-differential equation?

An integro-differential equation is a type of mathematical equation that involves both derivatives and integrals. It is used to model dynamic systems where the rate of change of a variable is dependent on the current value of that variable and on its past values.

2. How can I solve an integro-differential equation with Mathematica?

Mathematica has built-in functions for solving integro-differential equations, such as NDSolve and Integrate. These functions use numerical methods to find approximate solutions to the equation.

3. Can Mathematica handle nonlinear integro-differential equations?

Yes, Mathematica can handle both linear and nonlinear integro-differential equations. However, the methods for solving them may differ, and nonlinear equations may require more computational resources.

4. Is it possible to find an exact solution to an integro-differential equation?

In general, it is not always possible to find an exact solution to an integro-differential equation. However, Mathematica can symbolically manipulate the equation and provide a solution in terms of special functions, if one exists.

5. Are there any limitations to using Mathematica for integro-differential equations?

The main limitation of using Mathematica for integro-differential equations is the computational cost. Some equations may require significant amounts of time and memory to solve, especially if they are nonlinear or have complex boundary conditions.

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