### AIMS Mathematics

2021, Issue 6: 5737-5749. doi: 10.3934/math.2021338
Research article Special Issues

# Analytical solutions of $q$-fractional differential equations with proportional derivative

• Received: 06 November 2020 Accepted: 18 January 2021 Published: 26 March 2021
• MSC : 26A33, 39A13

• In this paper, we aim to propose a novel $q$-fractional derivative in the Caputo sense included proportional derivative. To this end, we firstly introduced a new concept of proportional $q$-derivative and discussed its properties in detail. Then, we add this definition in the concept of Caputo derivative to state a new type of dynamical system with $q$-calculus. For analytically solving this system, $q$-Laplace transform has been successfully applied to obtain the solutions. Indeed, the bivariate Mittag-Leffler function has an essential role in this regard. Two illustrative examples are also given in detail.

Citation: Aisha Abdullah Alderremy, Mahmoud Jafari Shah Belaghi, Khaled Mohammed Saad, Tofigh Allahviranloo, Ali Ahmadian, Shaban Aly, Soheil Salahshour. Analytical solutions of $q$-fractional differential equations with proportional derivative[J]. AIMS Mathematics, 2021, 6(6): 5737-5749. doi: 10.3934/math.2021338

### Related Papers:

• In this paper, we aim to propose a novel $q$-fractional derivative in the Caputo sense included proportional derivative. To this end, we firstly introduced a new concept of proportional $q$-derivative and discussed its properties in detail. Then, we add this definition in the concept of Caputo derivative to state a new type of dynamical system with $q$-calculus. For analytically solving this system, $q$-Laplace transform has been successfully applied to obtain the solutions. Indeed, the bivariate Mittag-Leffler function has an essential role in this regard. Two illustrative examples are also given in detail.

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