Pseudo-fractional differential equations and generalized g-Laplace transform

Area of Science:

• Mathematics
• Fractional Calculus
• Integral Transforms

Background:

• Integral transform theory is fundamental in solving differential equations.
• Pseudo-fractional derivatives extend classical fractional calculus concepts.
• The g-Laplace transform is a generalization with potential applications.

Purpose of the Study:

• To introduce a generalized g-Laplace transform.
• To explore its properties within integral transform theory.
• To investigate pseudo-fractional differential equations.

Main Methods:

• Development of a generalized g-Laplace transform.
• Analysis of integral transform properties.
• Application to pseudo-fractional differential equations involving a $\mathcal{H}^{\alpha,\beta}$ -Hilfer pseudo-fractional derivative and function convolution.

Main Results:

• The generalized g-Laplace transform is defined and its properties are discussed.
• The existence and uniqueness of solutions for a specific pseudo-fractional differential equation are investigated.
• Connections between the transform, pseudo-fractional derivatives, and function convolution are established.

Conclusions:

• The generalized g-Laplace transform provides a valuable tool for analyzing pseudo-fractional differential equations.
• The study confirms the existence and uniqueness of solutions for the investigated equation.
• This work contributes to the advancement of fractional calculus and integral transform theory.