Fractional Pantograph Research Articles

Involvement of three successive fractional derivatives in a system of pantograph equations and studying the existence solution and MLU stability

Abstract Developing a model of fractional differential systems and studying the existence and stability of a solution is considebly one of the most important topics in the field of analysis. Therefore, this manuscript was dedicated to deriving a new type of fractional system that arises from the combination of three sequential fractional derivatives with fractional pantograph equations. Also, the fixed-point technique was used to evaluate the existence and uniqueness of solutions to the supposed hybrid model. Furthermore, stability results for the intended system in the sense of the Mittag-Leffler-Ulam have been investigated. Ultimately, an illustrative example has been highlighted in order to reinforce the theoretical results and suggest applications for this article.

An explicit two-grid spectral deferred correction method for nonlinear fractional pantograph differential equations

An explicit two-grid spectral deferred correction method for nonlinear fractional pantograph differential equations

Numerical and analytical approach of nonlinear fractional pantograph nonlocal differential systems with non-singular kernel

Numerical and analytical approach of nonlinear fractional pantograph nonlocal differential systems with non-singular kernel

Qualitative Analysis for the Solutions of Fractional Stochastic Differential Equations

Fractional pantograph stochastic differential equations (FPSDEs) combine elements of fractional calculus, pantograph equations, and stochastic processes to model complex systems with memory effects, time delays, and random fluctuations. Ensuring the well-posedness of these equations is crucial as it guarantees meaningful, reliable, and applicable solutions across various disciplines. In differential equations, regularity refers to the smoothness of solution behavior. The averaging principle offers an approximation that balances complexity and simplicity. Our research contributes to establishing the well-posedness, regularity, and averaging principle of FPSDE solutions in Lp spaces with p≥2 under Caputo derivatives. The main ingredients in the proof include the use of Hölder, Burkholder–Davis–Gundy, Jensen, and Grönwall–Bellman inequalities, along with the interval translation approach. To understand the theoretical results, we provide numerical examples at the end.

Sparse wavelet Galerkin method: Application for fractional Pantograph problem

The multiwavelets Galerkin method is utilized to solve the fractional Pantograph equation. The method is based on rendering the desired problem into the equivalent integral equation with a weakly singular kernel (fractional integral equation). In this paper, for the first time, the wavelet properties are considered to solve and reduce the computational load for these equations. Compared to the use of other bases, even wavelets generated by a single generator, particularly Alpert’s multiwavelets, have been found to exhibit superior efficiency due to their unique properties. Our investigation supports the theoretical finding. The illustrative examples are also reported, demonstrating the accuracy and flexibility of the method.

Bernstein polynomials method for solving multi-order fractional neutral pantograph equations with error and stability analysis

In this investigation, we present a new method for addressing fractional neutral pantograph problems, utilizing the Bernstein polynomials method. We obtain solutions for the fractional pantograph equations by employing operational matrices of differentiation, derived from fractional derivatives in the Caputo sense applied to Bernstein polynomials. Error analysis, along with Chebyshev algorithms and interpolation nodes, is employed for solution characterization. Both theoretical and practical stability analyses of the method are provided. Demonstrative examples indicate that our proposed techniques occasionally yield exact solutions. We compare the algorithms using several established analytical methods. Our results reveal that our algorithm, based on Bernstein series solution methods, outperforms others, exhibiting superior performance with higher accuracy orders compared to those obtained from Chebyshev spectral methods, Bernoulli wavelet method, and Spectral Tau method.

Solvability and Ulam–Hyers–Rassias stability for generalized sequential quantum fractional pantograph equations

In the present manuscript, we discuss the existence, uniqueness an Ulam-stability of solutions for sequential fractional pantograph equations involving n Caputo and one Riemann–Liouville q−fractional derivatives. We prove the uniqueness of solutions for the given problem by using Banach’s contraction mapping principle. Then, the existence of at least one is obtained via Leray–Schauder’s alternative. Also, we define and study the Ulam-stability of solutions for the considered problem. Finally, an example is also given to point out the applicability of our main results.

Efficient spectral collocation method for nonlinear systems of fractional pantograph delay differential equations

<abstract><p>Caputo-Hadamard-type fractional calculus involves the logarithmic function of an arbitrary exponent as its convolutional kernel, which causes challenges in numerical approximations. In this paper, we construct and analyze a spectral collocation approach using mapped Jacobi functions as basis functions and construct an efficient algorithm to solve systems of fractional pantograph delay differential equations involving Caputo-Hadamard fractional derivatives. What we study is the error estimates of the derived method. In addition, we tabulate numerical results to support our theoretical analysis.</p></abstract>

Generalized Legendre Polynomial Configuration Method for Solving Numerical Solutions of Fractional Pantograph Delay Differential Equations

This paper develops a numerical approach for solving fractional pantograph delay differential equations using generalized Legendre polynomials. These polynomials are derived from generalized Taylor bases, which facilitate the approximation of the underlying analytical solutions, leading to the formulation of numerical solutions. The fractional pantograph delay differential equation is then transformed into a finite set of nonlinear algebraic equations using collocation points. Following this step, Newton's iterative method is applied to the resultant set of nonlinear algebraic equations to compute their numerical solutions. An error analysis for this methodology is subsequently presented, accompanied by numerical examples demonstrating its accuracy and efficiency. Overall, this study contributes a more streamlined and productive tool for determining the numerical solution of fractional differential equations.

On Caputo-Hadamard fractional pantograph problem of two different orders with Dirichlet boundary conditions

This manuscript aims to study the effectiveness of two different levels of fractional orders in the frame of Caputo-Hadamard (CH)-derivatives on a special type class of delay problem supplemented by Dirichlet boundary conditions. The corresponding Hadamard fractional integral equation is derived for a proposed CH-fractional pantograph system. The Banach, Schaefer, and Krasnoselskii fixed point theorems (FPTs), are used to investigate sufficient conditions of the existence and uniqueness theorems for the proposed system. Furthermore, the Green functions properties are investigated and used to discuss the Ulam-Hyers (UH) stability and its generalized by utilizing nonlinear analysis topics. Finally, three mathematical examples are provided with numerical results and figures by using Matlab software to illustrate the validity of our findings.

Large deviations for stochastic fractional pantograph differential equation

Large deviations for stochastic fractional pantograph differential equation

Fractional-order hybrid functions combining simulated annealing algorithm for solving fractional pantograph differential equations

A new and novel numerical method has been developed based on the fractional-order hybrid functions combining simulated annealing (SA) algorithm for the solutions of fractional pantograph delay differential equations. First of all, a fractional-order hybrid of block-pulse functions and Chebyshev polynomials (FOHBPCs) is defined. With the aid of regularized beta function, the exact formulas of FOHBPCs are derived under the definition of Riemann–Liouville fractional integral. And then, by the properties of FOHBPCs and the exact formulas together with the collocation method, the problem under consideration is simplified into algebraic equations, which are solved by Gaussian elimination method and Picard iteration method for linear and nonlinear cases, respectively. The error analysis of the proposed method is investigated. Due to the importance of the parameter α in the fractional-order hybrid functions method, SA algorithm is considered to find the optimal parameter α. Finally, the effectiveness and applicability of the suggested method are verified through some numerical examples.

Existence and Uniqueness Results for a Pantograph Boundary Value Problem Involving a Variable-Order Hadamard Fractional Derivative

This paper discusses the problem of the existence and uniqueness of solutions to the boundary value problem for the nonlinear fractional-order pantograph equation, using the fractional derivative of variable order of Hadamard type. The main results are proved through the application of fractional calculus and Krasnoselskii’s fixed-point theorem. Moreover, the Ulam–Hyers–Rassias stability of the nonlinear fractional pantograph equation is analyzed. To conclude this paper, we provide an example illustrating our findings and approach.

A spectral shifted gegenbauer collocation method for fractional pantograph partial differential equations and its error analysis

A spectral shifted gegenbauer collocation method for fractional pantograph partial differential equations and its error analysis

Efficient results on Hilfer pantograph model with nonlocal integral condition

The optimistic plan of this work reveals the solvability conditions of Hilfer fractional pantograph integrodifferential equations of type 0≤q2≤1 and order 0<q1<1. The presence of an integral solution and its uniqueness have been discussed with the ultimate fixed point techniques. At last, the observed results are substantiated by the illustration.

Fractional Pantograph Delay Equations Solving by the Meshless Methods

This work describes two efficient and useful methods for solving fractional pantograph delay equations (FPDEs) with initial and boundary conditions. These two methods depend mainly on orthogonal polynomials, which are the method of the operational matrix of fractional derivative that depends on Bernstein polynomials and the operational matrix of the fractional derivative with Shifted Legendre polynomials. The basic procedure of this method is to convert the pantograph delay equation to a system of linear equations and by using, the operational matrices we get rid of the integration and differentiation operations, which makes solving the problem easier. The concept of Caputo has been used to describe fractional derivatives. Finally, some numerical examples are identified to show the utility and capability of the two proposed approaches. Mathematica®12 program has been relied upon in the calculations.

A Generalized Lyapunov Inequality for a Pantograph Boundary Value Problem Involving a Variable Order Hadamard Fractional Derivative

The authors obtain existence and uniqueness results for a nonlinear fractional pantograph boundary value problem containing a variable order Hadamard fractional derivative. This type of model is appropriate for applications involving processes that occur in strongly anomalous media. They also derive a generalized Lyapunov-type inequality for the problem considered. Their results are obtained by the fractional calculus and Krasnosel’skii’s fixed point theorem. An example is given to illustrate their approach.

Fractional-Order Mittag–Leffler Functions for Solving Multi-dimensional Fractional Pantograph Delay Differential Equations

This manuscript is devoted to presenting an approximation method based upon a new set of fractional functions named fractional-order Mittag–Leffler functions (FM-LFs). This scheme is implemented to approximate the solution of a multi-dimensional fractional pantograph differential equation. To this approach, FM-LFs are introduced. Then, we employ FM-LFs to construct the pseudo-operational matrix of fractional integration and pantograph operational matrix (P-OM). We reduce the considered problems to systems of algebraic equations with the help of the mentioned matrices, and the collocation technique, respectively. Error analysis is proposed. Moreover, several numerical experiments have been considered to confirm the efficiency and applicability of the suggested scheme.

Nonlinear Piecewise Caputo Fractional Pantograph System with Respect to Another Function

The existence, uniqueness, and various forms of Ulam–Hyers (UH)-type stability results for nonlocal pantograph equations are developed and extended in this study within the frame of novel psi-piecewise Caputo fractional derivatives, which generalize the piecewise operators recently presented in the literature. The required results are proven using Banach’s contraction mapping and Krasnoselskii’s fixed-point theorem. Additionally, results pertaining to UH stability are obtained using traditional procedures of nonlinear functional analysis. Additionally, in light of our current findings, a more general challenge for the pantograph system is presented that includes problems similar to the one considered. We provide a pertinent example as an application to support the theoretical findings.

Ritz-generalized Pell wavelet method: Application for two classes of fractional pantograph problems

In this work, a computational method for solving fractional pantograph differential equations and fractional pantograph optimal control problems is proposed. The present technique is based on a new set of wavelet functions and a combination of Ritz and collocation methods. To this aim, we construct generalized Pell wavelets (GPws). An extra Caputo pseudo-operational matrix and pantograph operational matrix of GPws as new achievements are presented. To more easily calculate the fractional derivative of GPws, we define generalized piecewise Taylor functions (GPTfs). Then, utilizing these matrices, the Ritz method, and the collocation method, we find an approximate solution for each of the considered problems. An error analysis is proposed. Finally, some illustrative numerical tests are given to display the accuracy and effectiveness of the developed scheme.