A three dimensional problem can be approximated by either a two-dimensional or one-dimensional case, but some information will be lost. To reveal the lost information due to the lower dimensional approach, two-scale mathematics is needed. Generally one scale is established by usage where traditional calculus works, and the other scale is for revealing the lost information where the continuum assumption might be forbidden, and fractional calculus or fractal calculus has to be used. The two-scale transform can approximately convert the fractional calculus into its traditional partner, making the two-scale thermodynamics much promising.

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Electrospinning is a complex process, and it can be modeled by a Bratu-type equation with fractal derivatives by taking into account the solvent evaporation. Though there are many analytical methods available for such a problem, e.g. the variational iteration method and the homotopy perturbation method, a straightforward method with a simple solution process and high accurate results is much needed. This paper applies the Taylor series technology to fractal calculus, and an analytical approximate solution is obtained. A fractal variational principle is also discussed. As the Taylor series is accessible to all non-mathematicians, this paper sheds a bright light on practical applications of fractal calculus.

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Purpose The purpose of this paper is to demonstrate that the numerical method is not everything for nonlinear equations. Some properties cannot be revealed numerically; an example is used to elucidate the fact. Design/methodology/approach A variational principle is established for the generalized KdV – Burgers equation by the semi-inverse method, and the equation is solved analytically by the exp-function method, and some exact solutions are obtained, including blowup solutions and discontinuous solutions. The solution morphologies are studied by illustrations using different scales. Findings Solitary solution is the basic property of nonlinear wave equations. This paper finds some new properties of the KdV–Burgers equation, which have not been reported in open literature and cannot be effectively elucidated by numerical methods. When the solitary solution or the blowup solution is observed on a much small scale, their discontinuous property is first found. Originality/value The variational principle can explain the blowup and discontinuous properties of a nonlinear wave equation, and the exp-function method is a good candidate to reveal the solution properties.