Hyperasymptotic solutions for certain partial differential equations

We present the hyperasymptotic expansions for a certain group of solutions of the heat equation. We extend this result to a more general case of linear PDEs with constant coefficients. The generalisation is based on the method of Borel summability, which allows us to find integral representations of solutions for such PDEs.Comment: 17 page

On the computation of parameter derivatives of solutions of linear difference equations

AbstractA method is given to compute the parameter derivatives of recessive solutions of second-order inhomogeneous linear difference equations. The case of difference equations in which all solutions have the same rate of growth is also discussed.The method is illustrated by numerical computations of parameter derivatives of incomplete gamma functions and confluent hypergeometric functions

Asymptotic expansions for qq-gamma, qq-exponential, and qq-Bessel functions

AbstractNew asymptotic expansions are given for the q-gamma function, the q-exponential functions, and for the Hahn-Exton q-Bessel function. For the theta functions, four expansions are given. And for the Hahn-Exton q-Bessel difference equation, a new solution is given, which forms with the Hahn-Exton q-Bessel function a numerically satisfactory pair of solutions

Hyperasymptotics and hyperterminants: Exceptional cases

AbstractA new method is introduced for the computation of hyperterminants. It is based on recurrence relations, and can also be used to compute the parameter derivatives of the hyperterminants. These parameter derivatives are needed in hyperasymptotic expansions in exceptional cases. Numerical illustrations and an application are included

Exponentially accurate solution tracking for nonlinear ODEs, the higher order Stokes phenomenon and double transseries resummation

We demonstrate the conjunction of new exponential-asymptotic effects in the context of a second order nonlinear ordinary differential equation with a small parameter. First, we show how to use a hyperasymptotic, beyond-all-orders approach to seed a numerical solver of a nonlinear ordinary differential equation with sufficiently accurate initial data so as to track a specific solution in the presence of an attractor. Second, we demonstrate the necessary role of a higher order Stokes phenomenon in analytically tracking the transition between asymptotic behaviours in a heteroclinic solution. Third, we carry out a double resummation involving both subdominant and sub-subdominant transseries to achieve the two-dimensional (in terms of the arbitrary constants) uniform approximation that allows the exploration of the behaviour of a two parameter set of solutions across wide regions of the independent variable. This is the first time all three effects have been studied jointly in the context of an asymptotic treatment of a nonlinear ordinary differential equation with a parameter. This paper provides an exponential asymptotic algorithm for attacking such problems when they occur. The availability of explicit results would depend on the individual equation under study

Uniform asymptotic approximation of Fermi—Dirac integrals

AbstractThe Fermi–Dirac integral Fq(x)=1Γ(q+1)∫0∞tq1+et−xdt, q>−1, is considered for large positive values of x and q. The results are obtained from a contour integral in the complex plane. The approximation contains a finite sum of simple terms, an incomplete gamma function and an infinite asymptotic series. As follows from earlier results, the incomplete gamma function can be approximated in terms of an error function