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First, in joint work with S. Bodine of the University of Puget Sound, Tacoma, Washington, USA, we consider the second-order differential equation e2 y''=(1+e2 ψ(x, e))y with a small parameter e, where ψ is analytic and even with respect to e. It is well known that it has two formal solutions of the form y±(x,e)=e±x/eh±(x,e), where h±(x,e) is a formal series in powers of e whose coefficients are functions of x. It has been shown that one (resp. both) of these solutions are 1-summable in certain directions if ψ satisfies certain conditions, in particular concerning its x-domain. We show that these conditions are essentially necessary for 1-summability of one (resp. both) of the above formal solutions. In the proof, we solve a certain inverse problem: constructing a differential equation corresponding to a certain Stokes phenomenon. The second part of the paper presents joint work with Augustin Fruchard of the University of La Rochelle, France, concerning inverse problems for the general (analytic) linear equations e r y' = A(x,e) y in the neighborhood of a nonturning point and for second-order (analytic) equations e y'' - 2xy'-g(x,e) y=0 exhibiting resonance in the sense of Ackerberg-O'Malley, i.e., satisfying the Matkowsky condition: there exists a nontrivial formal solution $$\hat y\left( {x{\text{, }}\varepsilon } \right) = \sum {y_n } \left( x \right)\varepsilon ^n $$ such that the coefficients have no poles at x=0.

In recent years, the theory of Borel summability or multisummability of divergent power series of one variable has been established and it has been proved that every formal solution of an ordinary differential equation with irregular singular point is multisummable. For partial differential equations the summability problem for divergent solutions has not been studied so well, and in this paper we shall try to develop the Borel summability of divergent solutions of the Cauchy problem of the complex heat equation, since the heat equation is a typical and an important equation where we meet diveregent solutions. In conclusion, the Borel summability of a formal solution is characterized by an analytic continuation property together with its growth condition of Cauchy data to infinity along a stripe domain, and the Borel sum is nothing but the solution given by the integral expression by the heat kernel. We also give new ways to get the heat kernel from the Borel sum by taking a special Cauchy data.

SynopsisWe give a direct and simple proof of a central result of the above paper.