numeric-prelude-0.1: An experimental alternative hierarchy of numeric type classesSource codeContentsIndex
Lazy evaluation allows for the solution of differential equations in terms of power series. Whenever you can express the highest derivative of the solution as explicit expression of the lower derivatives where each coefficient of the solution series depends only on lower coefficients, the recursive algorithm will work.
solveDiffEq0 :: C a => [a]
verifyDiffEq0 :: C a => [a]
propDiffEq0 :: Bool
solveDiffEq1 :: (C a, C a) => [a]
verifyDiffEq1 :: (C a, C a) => [a]
propDiffEq1 :: Bool
solveDiffEq0 :: C a => [a]Source

Example for a linear equation: Setup a differential equation for y with

    y   t = (exp (-t)) * (sin t)
    y'  t = -(exp (-t)) * (sin t) + (exp (-t)) * (cos t)
    y'' t = -2 * (exp (-t)) * (cos t)

Thus the differential equation

    y'' = -2 * (y' + y)


The following function generates a power series for exp (-t) * sin t by solving the differential equation.

verifyDiffEq0 :: C a => [a]Source
propDiffEq0 :: BoolSource
solveDiffEq1 :: (C a, C a) => [a]Source
We are not restricted to linear equations! Let the solution be y with y t = (1-t)^-1 y' t = (1-t)^-2 y'' t = 2*(1-t)^-3 then it holds y'' = 2 * y' * y
verifyDiffEq1 :: (C a, C a) => [a]Source
propDiffEq1 :: BoolSource
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