| Summary: | In this article, we developed the idea of <i>q</i>-time scale calculus in quantum geometry. It includes the <i><i>q</i>-</i>time scale integral operators and<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo> </mo><msub><mo>∆</mo><mi>q</mi></msub></mrow></semantics></math></inline-formula>-differentials. It analyzes the fundamental principles which follow the calculus of <i>q</i>-time scales compared with the Leibnitz–Newton usual calculus and have few crucial consequences. The <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>∆</mo><mi>q</mi></msub></mrow></semantics></math></inline-formula>-differential reduced method of transformations was proposed to work out on partial <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="sans-serif">Δ</mi><mi>q</mi></msub></mrow></semantics></math></inline-formula>-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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>σ</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mi>t</mi></mrow></semantics></math></inline-formula> and <i>q</i> = 1; then, the solution merges with usual calculus for the mentioned initial value problem. The finding of the present work is that the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="sans-serif">Δ</mi><mi>q</mi></msub></mrow></semantics></math></inline-formula>-differential transformation reduced method is convenient and efficient.
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