Analytical solutions for autonomous differential equations with weighted derivatives

Abstract

In this work, we introduce a new definition of weighted derivatives along with corresponding integral operators, which aim to facilitate the solution of both linear and non-linear differential equations. A significant finding is that the fractional derivative of Caputo–Fabrizio type is a special case within this framework, allowing us to build upon existing research in this area. Additionally, we provide closed-form analytical solutions for autonomous and logistic equations using our newly defined derivatives and integrals. We thoroughly explore the properties associated with these weighted derivatives and integrals. To demonstrate the reliability and practical applicability of our results, we include several examples and applications that highlight the effectiveness of our approach.

Keywords

• Weighted derivatives
• Fractional derivative of Caputo–Fabrizio type
• Fractional calculus
• Fractional differential equations
• Exact differential equations