Application of Vieta–Lucas Series to Solve a Class of Multi-Pantograph Delay Differential Equations with Singularity

Izadi, Mohammad; Yüzbaşı, Şuayip; Ansari, Khursheed J.

doi:10.3390/sym13122370

Cited by 23 publications

(9 citation statements)

References 30 publications

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“…PDEs are widely used in probability theory, nonlinear dynamical systems, astrophysics, quantum mechanics, electrodynamics, and cell growth [61][62][63][64]. In this study, we consider the following FMPS:…”

Section: Introductionmentioning

confidence: 99%

A Novel Numerical Technique for Fractional Ordinary Differential Equations with Proportional Delay

Liaqat

Khan

Akgül

et al. 2022

Journal of Function Spaces

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Some researchers have combined two powerful techniques to establish a new method for solving fractional-order differential equations. In this study, we used a new combined technique, known as the Elzaki residual power series method (ERPSM), to offer approximate and exact solutions for fractional multipantograph systems (FMPS) and pantograph differential equations (PDEs). In Caputo logic, the fractional-order derivative operator is measured. The Elzaki transform method and the residual power series method (RPSM) are combined in this novel technique. The suggested technique is based on a new version of Taylor’s series that generates a convergent series as a solution. Establishing the coefficients for a series, like the RPSM, necessitates computing the fractional derivatives each time. As ERPSM just requires the concept of a zero limit, we simply need a few computations to get the coefficients. The novel technique solves nonlinear problems without the need for He’s and Adomian polynomials, which is an advantage over the other combined methods based on homotopy perturbation and Adomian decomposition methods. The relative, recurrence, and absolute errors of the problems are analyzed to evaluate the efficiency and consistency of the presented method. Graphical significances are also identified for various values of fractional-order derivatives. As a result, the procedure is quick, precise, and easy to implement, and it yields outstanding results.

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Section: Introductionmentioning

confidence: 99%

A Novel Numerical Technique for Fractional Ordinary Differential Equations with Proportional Delay

Liaqat

Khan

Akgül

et al. 2022

Journal of Function Spaces

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show abstract

“…Also, the authors of [13] applied a Besselbased technique for understanding the behavior of the solution of the multipantograph equation systems with mixed conditions. ere are other methods for analyzing the PD model such that including the exponential approximation technique [14], shifted Legendre method [15], backpropagated intelligent network technique [16], Gudermannian neural network [17], Hermite polynomial solutions [18], and Vieta-Lucas series method [19]. ese extensive works on the solutions of PD models open the door to exploit the wider applications of similar models of the reallife phenomenon.…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 2. If the assumed approximate solution to the problem ( 8) is represented by (19) and (22), then the discrete form of the Bernoulli system can be given in the following form:…”

mentioning

confidence: 99%

Application of a Novel Collocation Approach for Simulating a Class of Nonlinear Third-Order Lane–Emden Model

Adel

Sabir

Rezazadeh

et al. 2022

Mathematical Problems in Engineering

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The present study aims to design a mathematical system based on the Lane–Emden third-order pantograph differential model by using the general forms of the pantograph as well as the Lane–Emden models. The designed model is divided into two types along with the various singularity details at each point. The shape factors and the pantograph points are discussed for each type of the newly designed nonlinear third-order pantograph differential model. The Bernoulli collocation scheme is implemented to find the numerical results of the novel model. To show the reliability of the designed novel nonlinear model, four different variants have been solved. Moreover, the comparison of the obtained results with the exact solutions is presented to check the accuracy of the designed novel model.

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“…Utilizing diverse kinds of polynomials such as Legendre, Chebyshev, Bessel, Chelyshkov, Laguerre, and Vieta-Lucas functions is considered in literature. [14][15][16][17][18][19][20][21][22][23] Combinations of orthogonal functions with quasilinearization method (QLM) have been successfully applied to many important models in physical sciences, see, cf. previous studies.…”

Section: Introductionmentioning

confidence: 99%

“…It is well known that collocation‐based numerical approximations provide a promising tool to treat various initial and boundary value model problems in science and engineering. Utilizing diverse kinds of polynomials such as Legendre, Chebyshev, Bessel, Chelyshkov, Laguerre, and Vieta‐Lucas functions is considered in literature 14–23 . Combinations of orthogonal functions with quasilinearization method (QLM) have been successfully applied to many important models in physical sciences, see, cf.…”

Section: Introductionmentioning

confidence: 99%

A highly accurate matrix method for solving a class of strongly nonlinear BVP arising in modeling of human shape corneal

Abbasi

Izadi

Hosseini

2022

Math Methods in App Sciences

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This research study presents a novel highly accurate matrix approach for numerical treatments of a strongly nonlinear boundary value problem occurring in the modeling of the corneal shape of the human eye. Using the technique of quasilinearization, the nonlinear model is reduced into a sequence of linearized problems. Then, a spectral collocation procedure based on novel shifted Vieta‐Fibonacci (SVF) is applied to transform each subproblem into a linear algebraic system of equations. An upper bound for the error is provided, and convergence analysis of the SVF series solution in the weighted L2$$ {L}_2 $$ norm is discussed. The technique of error correction is used to improve the SVF polynomial solutions by means of the residual error function. Various numerical tests are provided to demonstrate the accuracy and efficiency of the presented collocation algorithm. The validation of the proposed approach is shown by comparison with available existing numerical solutions.

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Application of Vieta–Lucas Series to Solve a Class of Multi-Pantograph Delay Differential Equations with Singularity

Cited by 23 publications

References 30 publications

A Novel Numerical Technique for Fractional Ordinary Differential Equations with Proportional Delay

A Novel Numerical Technique for Fractional Ordinary Differential Equations with Proportional Delay

Application of a Novel Collocation Approach for Simulating a Class of Nonlinear Third-Order Lane–Emden Model

A highly accurate matrix method for solving a class of strongly nonlinear BVP arising in modeling of human shape corneal

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