The method of asymptotic expansions is useful for obtaining an approximation for some function, call it f(x,epsilon), when we have enough information about f to calculate:
1. The lowest-order asymptotic limit of f(x,epsilon) as epsilon goes to zero.
2. A recurrence relation (derived from the method of asymptotic expansions) that calculates the higher-order terms from the lower-order terms.
Suppose I have that information about f for two limiting cases. One is the limit stated above in which epsilon goes to zero. The other is the opposite limit in which epsilon goes to +infinity (1/epsilon goes to zero from above). Is there a theory, that has some resemblance to the method of asymptotic expansions, that can utilize all of this information to interpolate between the two limiting cases?
I should say a little more about my application before being advised to use matched asymptotic expansions. The method of matched asymptotic expansions is useful when f(x,epsilon) is governed by a differential equation in the x-variable and we can partition the relevant x-interval into two sections, with the small-epsilon approximation useful for one section, and the large-epsilon approximation useful for the other section. The boundary value of f between the sections is adjustable and is adjusted to make the two solutions join together (match). In my application, the small-epsilon approximation applies to the entire x-interval when it applies, and the large-epsilon approximation applies to the entire x-interval when it applies. I have no adjustable parameters that can make the two limits join together. So I am looking for what might be a new theory (new to me but maybe well-known by some others). The theory is all about an approximate interpolation between two limits while becoming exact at both limits.
Perhaps Two-point Pade approximation might be helpful, e.g. https://doi.org/10.1016/0771-050X(80)90012-1.
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