Converting IVP to Volterra equations

In mathematical analysis, converting an Initial Value Problem (IVP) for a differential equation into an equivalent Volterra Integral Equation is a standard approach. This transformation is crucial in numerical methods, as integral equations can sometimes be easier to handle computationally.

Mathematical Background

Initial Value Problem (IVP)

An IVP typically involves solving a differential equation along with initial conditions. For example:

where:

• $y(x)$ is the unknown function to be solved,
• $f(x, y(x))$ is a given function,
• $x_0$ is the initial point, and
• $y_0$ is the initial value.

Volterra Integral Equation

A Volterra integral equation of the second kind has the general form:

where:

• $f(x)$ is a given function,
• $K(x, t)$ is the kernel,
• $y(t)$ is the unknown function.

Procedure for Conversion

Step 1: Rewrite the Differential Equation

Start with the first-order IVP:

Step 2: Integrate Both Sides

Integrate the differential equation from $x_0$ to $x$:

The left-hand side becomes:

Thus:

Step 3: Express as a Volterra Integral Equation

The result is a Volterra integral equation of the second kind:

Here:

• $f(t, y(t))$ plays the role of the kernel $K(x, t)$,
• $y_0$ is the initial value.

Example 1: Linear IVP

Problem:

Solve the IVP:

Conversion:

1. Integrate both sides:

   $$y(x) = y(0) + \int_0^x [-2y(t) + 4t] \, dt$$

2. Substitute $y(0) = 1$:

   $$y(x) = 1 + \int_0^x [-2y(t) + 4t] \, dt$$

3. Split the integral:

   $$y(x) = 1 – 2 \int_0^x y(t) \, dt + \int_0^x 4t \, dt$$

4. Evaluate the second term:

   $$\int_0^x 4t \, dt = [2t^2]_0^x = 2x^2$$

Thus, the Volterra equation is:

Example 2: Nonlinear IVP

Problem:

Solve the IVP:

Conversion:

1. Integrate both sides:

   $$y(x) = y(0) + \int_0^x [y^2(t) + e^t] \, dt$$

2. Substitute $y(0) = 0$:

   $$y(x) = \int_0^x y^2(t) \, dt + \int_0^x e^t \, dt$$

3. Simplify:

   $$y(x) = \int_0^x y^2(t) \, dt + [e^t]_0^x = \int_0^x y^2(t) \, dt + e^x – 1$$

Thus, the Volterra equation is:

Numerical and Computational Considerations

• Picard Iteration:
  Iterative methods like Picard iteration can be applied to solve the resulting Volterra integral equation numerically. Start with an initial guess $y_0(x)$ and refine it iteratively:

  $$y_{n+1}(x) = y_0 + \int_{x_0}^x f(t, y_n(t)) \, dt$$

• Trapezoidal Rule:
  For numerical integration, apply the trapezoidal rule or other quadrature methods:

  $$\int_{x_0}^x f(t, y(t)) \, dt \approx \sum_{i=0}^{n} w_i f(t_i, y(t_i))$$

  where $w_i$ are weights.

References

1. Books

   • Tricomi, F.G. Integral Equations. Dover Publications, 1985.
   • Kanwal, Ram P. Linear Integral Equations: Theory and Technique. Birkhäuser, 1996.
   • Kreyszig, Erwin. Advanced Engineering Mathematics. Wiley, 2011.

2. Articles

   • “Integral Equation Methods in Differential Equations,” Mathematics of Computation, 1972.
   • “Numerical Techniques for Solving Volterra Equations,” SIAM Journal on Numerical Analysis, 1985.

3. Online Resources

   • MIT OpenCourseWare: https://ocw.mit.edu (Search for integral equations and differential equations topics).
   • Wolfram MathWorld: Volterra Integral Equations.