Search

The extension of the theory of generalized fractal–fractional calculus, named in this article as Martínez–Kaabar Fractal–Fractional (MKFF) calculus, is addressed to the field of integral equations. Based on the classic Adomian decomposition method, by incorporating the MKFF α , γ -integral operator, we establish the so-called extended Adomian decomposition method (EADM). The convergence of this proposed technique is also discussed. Finally, some interesting Volterra Integral equations of non-integer order which possess a fractal effect are solved via our proposed approach. The results in this work provide a novel approach that can be employed in solving various problems in science and engineering, which can overcome the challenges of solving various equations, formulated via other classical fractional operators. [ABSTRACT FROM AUTHOR]

In this study, a new generalized fractal–fractional (FF) derivative is proposed. By applying this definition to some elementary functions, we show its compatibility with the results of the FF derivative in the Caputo sense with the power law. The main elements of classical differential calculus are introduced in terms of this new derivative. Thus, we establish and demonstrate the basic operations with derivatives, chain rule, mean value theorems with their immediate applications and inverse function's derivative. We complete the theory of generalized FF calculus by proposing a notion of integration and presenting two important results of integral calculus: the fundamental theorem and Barrow's rule. Finally, we analytically solve interesting FF ordinary differential equations by applying our proposed definition. [ABSTRACT FROM AUTHOR]

The extension of the theory of generalized fractal–fractional calculus, named in this article as Martínez–Kaabar Fractal–Fractional (MKFF) calculus, is addressed to the field of integral equations. Based on the classic Adomian decomposition method, by incorporating the MKFF α,γ-integral operator, we establish the so-called extended Adomian decomposition method (EADM). The convergence of this proposed technique is also discussed. Finally, some interesting Volterra Integral equations of non-integer order which possess a fractal effect are solved via our proposed approach. The results in this work provide a novel approach that can be employed in solving various problems in science and engineering, which can overcome the challenges of solving various equations, formulated via other classical fractional operators.

In this study, a new generalized fractal–fractional (FF) derivative is proposed. By applying this definition to some elementary functions, we show its compatibility with the results of the FF derivative in the Caputo sense with the power law. The main elements of classical differential calculus are introduced in terms of this new derivative. Thus, we establish and demonstrate the basic operations with derivatives, chain rule, mean value theorems with their immediate applications and inverse function’s derivative. We complete the theory of generalized FF calculus by proposing a notion of integration and presenting two important results of integral calculus: the fundamental theorem and Barrow’s rule. Finally, we analytically solve interesting FF ordinary differential equations by applying our proposed definition.