Solve various integral equations using numerical methods.

# Solve Volterra and Fredholm integral equations

This Python package estimates Volterra and Fredholm integral equations using known techniques.

## Volterra This package provides the function SolveVolterra which approximates the solution, g(x), to the Volterra Integral Equation of the first kind: using the method in Betto and Thomas (2021).

See the "Trapezoid and Midpoint Rules" section for a discussion of these two rules.

### Parameters

k : function
The kernel function that takes two arguments.
f : function
The left hand side (free) function with f(a) = 0.
a : float
Lower bound of the integral, defaults to 0.
b : float
Upper bound of the estimate, defaults to 1.
num : int
Number of estimation points between zero and b.
method : string
Use either the 'midpoint' (default) or 'trapezoid' rule.


### Returns

grid : 2-D array
Input values are in the first row and output values are in the second row.


## Fredholm This package provides the function SolveFredholm which approximates the solution, g(x), to the Fredholm Integral Equation of the first kind: using the method described in Twomey (1963). It will return a smooth curve that is an approximate solution. However, it may not be a good approximate to the true solution.

### Parameters

k : function
The kernel function that takes two arguments.
f : function
The left hand side (free) function that takes one argument.
a : float
Lower bound of the of the Fredholm definite integral, defaults to -1.
b : float
Upper bound of the of the Fredholm definite integral, defaults to 1.
num : int
Number of estimation points between zero and b.
smin : float
Optional. Lower bound of enforcement values for s.
smax : float
Optional. Upper bound of enforcement values for s.
snum : int
Optional. Number of enforcement points for s.


### Returns

grid : 2-D array
Input values are in the first row and output values are in the second row.


# Trapezoid and Midpoint Rules

Volterra integral equations are typically solved using the midpoint rule. However, the trapezoid rule often converges faster. See below an example of the trapezoid rule performing well with just six grid points. Thus, the trapezoid rule typically performs better. However, the trapezoid rule is less stable than the midpoint rule. An example where this this instability is an issue is provided below. This can be remedied by smoothing the function. For example, with inteq.helpers.smooth().

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