Solving two simultaneous integro-differential equations

In summary, the conversation discusses the attempt to find a closed-form solution for two simultaneous integro-differential equations. The equations are given with initial conditions and involve positive constants. It is mentioned that an analytical expression for the equations is unlikely to be found, but a solution using numerical methods is possible. The conversation ends with a note of gratitude to the friends who provided help and the mention of using Mathematica 11 for the solution.
  • #1
Boudy
7
1
I am trying to find a closed-form (analytical) solution for the two following simultaneous integro-differential equations :

du[x]/dx= - a v[x] +b ∫〖[1-(y-x)^4 〗].(v[y]-v[x])dy
And
(dv[x])/dx= - f u[x] -g ∫〖[1-(y-x)^4 〗].u[y]dy
With the initial conditions:
v[0]=e and u[1]=0
a,b,f,g and e are positive constants
Both integrals are from y=0 to y=1.
The unknowns are u[x] and v[x].
 
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  • #2
an analytical expression for that set of equations is most likely not going to be found.
 
  • #3
Just wrote down the start to a solution.
Differentiate one of the equations with respect to x. You already have expressions for du/dx and dv/dx, so when you differentiate you can substitute the other equation in. Voila! Suddenly you've got an integro-differential equation for only one function. I'll leave you to do the rest of it ;)
 
  • #4
I thank both friends for their kind help. Fortunately, and after some modifications in the equations, a solution was possible using the numerical solution of two simultaneous differential equations in two variables and one single independent variable. This was done using the NDSolve command within a Mathematica 11 code.
Thanks again.
 

Related to Solving two simultaneous integro-differential equations

1. What is the definition of a simultaneous integro-differential equation?

A simultaneous integro-differential equation is a type of mathematical equation that involves both derivatives and integrals, with multiple variables and functions being solved simultaneously. It is commonly used in physics, engineering, and other fields to model complex systems.

2. How do you solve a system of two simultaneous integro-differential equations?

To solve a system of two simultaneous integro-differential equations, you can use a variety of techniques such as separation of variables, Laplace transforms, or numerical methods. The specific method used will depend on the complexity of the equations and the desired level of accuracy.

3. What are the applications of solving two simultaneous integro-differential equations?

Solving two simultaneous integro-differential equations has many practical applications, including modeling of physical systems such as fluid dynamics, heat transfer, and electrical circuits. It is also used in economics, biology, and other fields to analyze complex systems and make predictions.

4. What are some common challenges in solving two simultaneous integro-differential equations?

One of the main challenges in solving two simultaneous integro-differential equations is the complexity of the equations themselves. They can be difficult to solve analytically and may require the use of numerical methods. Additionally, accurately modeling real-world systems can be challenging and may require simplifications or assumptions.

5. Are there any software or tools available to help solve two simultaneous integro-differential equations?

Yes, there are many software and tools available to help solve two simultaneous integro-differential equations, such as MATLAB, Mathematica, and Maple. These programs have built-in functions and algorithms specifically designed for solving complex mathematical equations, making the process more efficient and accurate.

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