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Solve various integral equations using numerical methods.

Project description

Solve Volterra and Fredholm integral equations

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

Volterra

plot generated by package

This package provides the function SolveVolterra which approximates the solution, g(x), to the Volterra Integral Equation of the first kind:

f(s) = \int_a^s K(s,y) g(y) dy

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

Fredholm plot generated by package

This package provides the function SolveFredholm which approximates the solution, g(x), to the Fredholm Integral Equation of the first kind:

f(s) = \int_a^b K(s,y) g(y) dy

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.

example of trapezoid rule converging faster

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.

example of trapezoid rule having issues

This can be remedied by smoothing the function. For example, with inteq.helpers.smooth().

Project details


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