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. An early example of the dominant balance method is the Newton polygon method. 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. 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.
Asymptotic relations The functions f ( z ) {\textstyle f(z)} and g ( z ) {\displaystyle g(z)} of parameter or independent variable z {\textstyle z} and the quotient f ( z ) / g ( z ) {\textstyle f(z)/g(z)} have limits as z {\textstyle z} approaches the limit L {\textstyle L} . The function f ( z ) {\textstyle f(z)} is much less than g ( z ) {\textstyle g(z)} as z {\textstyle z} approaches L {\textstyle L} , written as f ( z ) ≪ g ( z ) ( z → L ) {\textstyle f(z)\ll g(z)\ (z\to L)} , if the limit of the quotient f ( z ) / g ( z ) {\textstyle f(z)/g(z)} is zero as z {\textstyle z} approaches L {\textstyle L} . The relation f ( z ) {\textstyle f(z)} is lower order than g ( z ) {\textstyle g(z)} as z {\textstyle z} approaches L {\textstyle L} , written using little-o notation f ( z ) = o ( g ( z ) ) ( z → L ) {\textstyle f(z)=o(g(z))\ (z\to L)} , is identical to the f ( z ) {\textstyle f(z)} is much less than g ( z ) {\textstyle g(z)} as z {\textstyle z} approaches L {\textstyle L} relation. The function f ( z ) {\textstyle f(z)} is equivalent to g ( z ) {\textstyle g(z)} as z {\textstyle z} approaches L {\textstyle L} , written as f ( z ) ∼ g ( z ) ( z → L ) {\textstyle f(z)\sim g(z)\ (z\to L)} , if the limit of the quotient f ( z ) / g ( z ) {\textstyle f(z)/g(z)} is 1 as z {\textstyle z} approaches L {\textstyle L} . This result indicates that the zero function, f ( z ) = 0 {\textstyle f(z)=0} for all values of z {\textstyle z} , can never be equivalent to any other function. Asymptotically equivalent functions remain asymptotically equivalent under integration if requirements related to convergence are met. There are more specific requirements for asymptotically equivalent functions to remain asymptotically equivalent under differentiation.
Equation properties An equation's approximate solution is s ( z ) {\textstyle s(z)} as z {\textstyle z} approaches limit L {\textstyle L} . The equation's terms that may be constants or contain this solution are T 0 ( s ) , T 1 ( s ) , … , T n ( s ) {\textstyle T_{0}(s),T_{1}(s),\ldots ,T_{n}(s)} . If the approximate solution is fully correct, the equation's terms sum to zero in this equation:
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