AsymptoticDSolveValue

AsymptoticDSolveValue[eqn,f,{x,x0,n}]

computes an asymptotic approximation to the differential equation eqn for f[x] centered at x0 of order n.

AsymptoticDSolveValue[{eqn1,eqn2,},{f1,f2,},{x,x0,n}]

computes an asymptotic approximation to a system of differential equations.

AsymptoticDSolveValue[eqn,f,x,{ϵ,ϵ0,n}]

computes an asymptotic approximation of f[x,ϵ] for the parameter ϵ centered at ϵ0 of order n.

Details and Options

  • Asymptotic approximations to differential equations are also known as asymptotic expansions, perturbation solutions, regular perturbations and singular perturbations, etc. They are also known by specific methods used to compute some of them, such as Frobenius series, WKB, boundary-layer method, etc.
  • Asymptotic approximations are typically used to solve problems for which no exact solution can be found or to get simpler answers for computation, comparison and interpretation.
  • If the exact result is g[x] and the asymptotic approximation of order n at x0 is gn[x], then the result is AsymptoticLess[g[x]-gn[x],gn[x]-gn-1[x],xx0] or g[x]-gn[x]o[gn[x]-gn-1[x]] as xx0.
  • The asymptotic approximation gn[x] is often given as a sum gn[x]αkϕk[x], where {ϕ1[x],,ϕn[x]} is an asymptotic scale ϕ1[x]ϕ2[x]>ϕn[x] as xx0. Then the result is AsymptoticLess[g[x]-gn[x],ϕn[x],xx0] or g[x]-gn[x]o[ϕn[x]] as xx0.
  • Common asymptotic scales include:
  • Taylor scale when xx0
    Laurent scale when xx0
    Laurent scale when x±
    Puiseaux scale when xx0
  • The scales used to express the asymptotic approximation are automatically inferred from the problem and can often include more exotic scales.
  • The center x0 can be any finite or infinite real or complex number.
  • The order n must be a positive integer and specifies order of approximation for the asymptotic solution. It is not related to polynomial degree.
  • The following options can be given:
  • Assumptions$Assumptionsassumptions to make about parameters
    GenerateConditionsFalsewhether to generate answers that involve conditions on parameters
    GeneratedParametersChow to name generated parameters
    MethodAutomaticmethod to use
    PerformanceGoal"Quality"aspects of performance to optimize
  • Possible settings for PerformanceGoal include $PerformanceGoal, "Quality" and "Speed". With the "Quality" setting, AsymptoticDSolve typically solves more problems or produces simpler results, but it potentially uses more time and memory.

Examples

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Basic Examples  (2)

Find a series solution for a differential equation:

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Plot the solution:

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Find an asymptotic expansion for a perturbation problem:

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Plot the solution:

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Scope  (44)

Options  (1)

Applications  (7)

Properties & Relations  (3)

Possible Issues  (1)

See Also

DSolveValue  AsymptoticLess  Series  NDSolveValue  ParametricNDSolveValue  AsymptoticIntegrate

Introduced in 2018
(11.3)