Transformation function to turn FDEs into ODEs #3776
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test/fractional_to_ordinary.jl
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| sys = fractional_to_ordinary(eqs, x, α, 10^-7, 1) | ||
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| prob = ODEProblem(sys, [], tspan) | ||
| sol = solve(prob, radau5(), abstol = 1e-10, reltol = 1e-10) |
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| sol = solve(prob, radau5(), abstol = 1e-10, reltol = 1e-10) | |
| sol = solve(prob, radau5(), saveat=time, abstol = 1e-10, reltol = 1e-10) |
| rhs += value * t^(index - 1) / gamma(index) | ||
| end | ||
| for index in range(M, N-1; step=1) | ||
| new_z = Symbol(:z, :_, i) |
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Use a unicode piece so this doesn't clash with a user chosen name
| ``` | ||
| """ | ||
| function fractional_to_ordinary(eqs, variables, alphas, epsilon, T; initials = 0) | ||
| @independent_variables t |
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This should just be the same as the main t. @AayushSabharwal what's the quick way to check the eqs all match?
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That they have the same indepvar? check_equations(eqs, iv).
| alpha_0 = α - m + 1 | ||
| end | ||
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| δ = (gamma(alpha_0+1) * epsilon)^(1/alpha_0) |
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Gamma should be the gamma function from SpecialFunctions.
| sol = solve(prob, radau5(), abstol = 1e-5, reltol = 1e-5) | ||
| ``` | ||
| """ | ||
| function linear_fractional_to_ordinary(degrees, coeffs, rhs, epsilon, T; initials = 0) |
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how does this differ from just defining D.(u, alpha) ~ A*u and calling the first one?
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This is made for FDE with multiple fractional terms in a single equation, which the first one can't handle. On the other hand, this one doesn't work for non-linear DE.
Created a function for transformation addressed in #3707 using the methods described in https://arxiv.org/pdf/2506.04188.
Works for the first two examples of numerical experiments, but doesn't work for simple cases like x^2.