Integral transform approach to mimetic discrete calculus

We introduce an integral transform that maps differential equations and special functions of standard continuous calculus onto finite difference equations and deformed special functions of mimetic discrete calculus, or h-calculus. We show that our procedure leads to insightful reformulations of several problems in mathematics and physics where discrete equations play a significant role, such as in solving finite difference equations, in applying discrete versions of integral transforms, such as the h-Laplace transform, in solving master equations of stochastic physics, in developing a discrete version of H theory of multiscale complex hierarchical phenomena and in finding lattice Green's functions for describing quantum charge transport through phase coherent systems. We believe that our integral transform technique, or mimetic map, will help systematize the connections through analogy between discrete calculus and standard continuous calculus.

Automated summary

This paper introduces an integral transform technique that maps differential equations and special functions of standard continuous calculus onto finite difference equations and deformed special functions of mimetic discrete calculus. The technique has insightful applications in mathematics and physics, particularly in solving finite difference equations, applying discrete versions of integral transforms, solving master equations of stochastic physics, developing a discrete version of H theory, and finding lattice Green's functions for quantum charge transport.