In mathematics, the method of dominant balance approximates the solution to an equation by solving a simplified form of the equation containing 2 or more of the equation's terms that most influence (dominate) the solution and excluding terms contributing only small modifications to this approximate solution.
Following an initial solution, iteration of the procedure may generate additional terms of an asymptotic expansion providing a more accurate solution.
Newton developed this method to find an explicit approximation for an algebraic function.
Newton expressed the function as proportional to the independent variable raised to a power, retained only the lowest-degree polynomial terms (dominant terms), and solved this simplified reduced equation to obtain an approximate solution.
[3][4] Dominant balance has a broad range of applications, solving differential equations arising in fluid mechanics, plasma physics, turbulence, combustion, nonlinear optics, geophysical fluid dynamics, and neuroscience.
[7] Asymptotically equivalent functions remain asymptotically equivalent under integration if requirements related to convergence are met.
means make these terms equal and asymptotically equivalent by finding the function
[10][11] A consistent solution that balances two equation terms may generate an accurate approximation to the full equation's solution for
may generate simplified reduced equations for distinct exponent values of
[9] These simplified equations are called distinguished limits and identify balanced dominant equation terms.
The dominant balance method applies scale transformations to balance equation terms whose factors contain distinct exponents.
Scaled functions are applied to differential equations when
is an equation parameter, not the differential equation´s independent variable.
[5] The Kruskal-Newton diagram facilitates identifying the required scaled functions needed for dominant balance of algebraic and differential equations.
[5] For differential equation solutions containing an irregular singularity, the leading behavior is the first term of an asymptotic series solution that remains when the independent variable
The controlling factor is the fastest changing part of the leading behavior.
For each pair of distinct equation terms
the algorithm applies a scale transformation if needed, balances the selected terms by finding a function that solves the reduced equation and then determines if this function is consistent.
The process is repeated for each pair of distinct equation terms.
The method may be iterated to generate additional terms of an asymptotic expansion to provide a more accurate solution.
[5] The dominant balance method will find an explicit approximate expression for the multi-valued function
The set of approximate solutions has 5 functions:
The dominant balance method will find an approximate solution as
The set of approximate solutions has 2 functions:[10]
For other term pairs, the functions that solve the reduced equations are not consistent.
[10] The set of approximate solutions has 2 functions:[10]
and this means that a power series expansion can represent the remainder of the solution.
[10] The dominant balance method generates the leading term to this asymptotic expansion with constant
and expansion coefficients determined by substitution into the full differential equation:[10] A partial sum of this non-convergent series generates an approximate solution.
The leading term corresponds to the Liouville-Green (LG) or Wentzel–Kramers–Brillouin (WKB) approximation.