Existence and $\delta $-Approximate solution of implicit fractional pantograph equations in the frame of Hilfer-Katugampola operator

The major goal of this research paper is to investigate the existence and uniqueness of an implicit fractional pantograph equation in the frame of the Hilfer-Katugampola operator on the finite interval $[a,b]$ with mixed nonlocal conditions. Our analysis of the existence and uniqueness of solutions depends on some fixed point theorems such as Banach and Krasnoselskii. Moreover, we discuss the dependence of solutions on mixed nonlocal conditions by means of $\delta $-approximated solution. As an application, we provide an example to illustrate the validity of our results.

Asymptotic Stability of Nonlinear Discrete Fractional Pantograph Equations with Non-Local Initial Conditions

Pantograph, the technological successor of trolley poles, is an overhead current collector of electric bus, electric trains, and trams. In this work, we consider the discrete fractional pantograph equation of the form Δ*β[k](t)=wt+β,k(t+β),k(λ(t+β)), with condition k(0)=p[k] for t∈N1−β, 0<β≤1, λ∈(0,1) and investigate the properties of asymptotic stability of solutions. We will prove the main results by the aid of Krasnoselskii’s and generalized Banach fixed point theorems. Examples involving algorithms and illustrated graphs are presented to demonstrate the validity of our theoretical findings.

Dynamics and stability of ψ-fractional pantograph equations with boundary conditions

This manuscript is devoted to obtain some adequate conditions for existence of at least one solution to fractional pantograph equation (FPE) involving the ψ -fractional derivative. The proposed problem is studied under some boundary conditions. Since stability is an important aspect of the qualitative theory. Therefore, we also discuss the Ulam-Hyers and Ulam-Hyers-Rassias type stabilites for the considered problem. Our results are based on some standard fixed point theorems. For the demonstration of our results, we provide an example.

A Nonlinear Fractional Problem with Mixed Volterra-Fredholm Integro-Differential Equation: Existence, Uniqueness, H-U-R Stability, and Regularity of Solutions

This paper considers nonlinear fractional mixed Volterra-Fredholm integro-differential equation with a nonlocal initial condition. We propose a fixed-point approach to investigate the existence, uniqueness, and Hyers-Ulam-Rassias stability of solutions. Results of this paper are based on nonstandard assumptions and hypothesis and provide a supplementary result concerning the regularity of solutions. We show and illustrate the wide validity field of our findings by an example of problem with nonlocal neutral pantograph equation, involving functional derivative and ψ -Caputo fractional derivative.

Solution of Pantograph Equation by Collocation method using Orthogonal Exponential Polynomials(OEP)

Novel matrix based numerical technique known as collocation method is implemented for the solution of pantograph differential equations (PDE) via truncated orthoexponential polynomial(OEP). To check applicability, reliability and efficiency of the methodology, here examine three examples of delay differential equations. At last the comparison made between proposed and reported methodologies and present method was perfect in agreement.