Applications of (<i>h</i>,<i>q</i>)-Time Scale Calculus to the Solution of Partial Differential Equations <xref rid="fn1-csmf-2463464" ref-type="fn">†</xref>

Abstract

In this article, we developed the idea of q-time scale calculus in quantum geometry. It includes the q-time scale integral operators and ∆q-differentials. It analyzes the fundamental principles which follow the calculus of q-time scales compared with the Leibnitz–Newton usual calculus and have few crucial consequences. The ∆q-differential reduced method of transformations was proposed to work out on partial Δq-differential equations in time scale. With easily computable coefficients, the result is calculated in the version of a power series which is convergent. The performance and effectiveness of the proposed procedure are also illustrated, and Matlab software is applied for calculation with the support of some fascinating examples. It changes when σt=t and q = 1; then, the solution merges with usual calculus for the mentioned initial value problem. The finding of the present work is that the Δq-differential transformation reduced method is convenient and efficient.

Keywords

• Δ<i>q</i>-differential
• <i><i>q</i>-</i>time scale
• <i>q</i>-Integral operators
• Δ<i>q</i>-differential reduced transform method
• partial differential equations