Analysis of Impulsive Boundary Value Pantograph Problems via Caputo Proportional Fractional Derivative under Mittag–Leffler Functions

Abstract

This manuscript investigates an extended boundary value problem for a fractional pantograph differential equation with instantaneous impulses under the Caputo proportional fractional derivative with respect to another function. The solution of the proposed problem is obtained using Mittag–Leffler functions. The existence and uniqueness results of the proposed problem are established by combining the well-known fixed point theorems of Banach and Krasnoselskii with nonlinear functional techniques. In addition, numerical examples are presented to demonstrate our theoretical analysis.

Qualitative study for impulsive pantograph fractional integro-differential equation via ψ-Hilfer derivative

“…φ-Caputo differential inclusion boundary value problems are studied in [7] supplemented with mixed integro-derivative conditions in the frame of the φ-Riemann-Liouville operators. Nonlinear impulsive pantograph fractional BVPs under Caputo proportional fractional derivative are investigated in [17]. The significance of our impulsive and initial conditions in (P) compare with the above results relies on the fact that they are nonlocal.…”

On Stability of Second Order Pantograph Fractional Differential Equations in Weighted Banach Space

“…Existence and uniqueness results for nonlinear neutral pantograph equations with generalized fractional derivative were proved in [14]. Several significant conclusions have been made on this subject using different iterations of the pantograph equation and various types of fractional operators [15][16][17][18][19][20]. Fixed point theorems are frequently used to demonstrate the solutions' existence and uniqueness.…”

A Novel Numerical Approach in Solving Fractional Neutral Pantograph Equations via the ARA Integral Transform

“…For example, the Adomian decomposition method (ADM) [13] was applied to solve nonlinear fractional diffusion and wave equations [14,15], and the homotopy perturbation method (HPM) was constructed in [16] and was used to study dissipative nonplanar solitons in an electronegative complex plasma [17], etc. [18][19][20][21][22][23]. Nowadays the residual power series method (RPSM) is used to construct power series solutions of differential equations without linearization, discretization, or perturbation [24][25][26].…”