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20 pages, 904 KB

Open AccessArticle

Analytical and Numerical Approaches via Quadratic Integral Equations

by Jihan Alahmadi, Mohamed A. Abdou and Mohamed A. Abdel-Aty

Axioms 2024, 13(9), 621; https://doi.org/10.3390/axioms13090621 - 12 Sep 2024

Cited by 1 | Viewed by 929

A quadratic integral Equation (QIE) of the second kind with continuous kernels is solved in the space

C ([0, T] \times [0, T]) .

The existence of at least one solution to the QIE is [...] Read more.

A quadratic integral Equation (QIE) of the second kind with continuous kernels is solved in the space

C ([0, T] \times [0, T]) .

The existence of at least one solution to the QIE is discussed in this article. Our evidence depends on a suitable combination of the measures of the noncompactness approach and the fixed-point principle of Darbo. The quadratic integral equation can be used to derive a system of integral equations of the second kind using the quadrature method. With the aid of two different polynomials, Laguerre and Hermite, the system of integral equations is solved using the collocation method. In each numerical approach, the estimation of the error is discussed. Finally, using some examples, the accuracy and scalability of the proposed method are demonstrated along with comparisons. Mathematica 11 was used to obtain all of the results from the techniques that were shown. Full article

(This article belongs to the Special Issue Applied Mathematics and Numerical Analysis: Theory and Applications)

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11 pages, 563 KB

Open AccessArticle

An Efficient Numerical Scheme for Fractional Order Mathematical Model of Cytosolic Calcium Ion in Astrocytes

by Devendra Kumar, Hunney Nama, Jagdev Singh and Jitendra Kumar

Fractal Fract. 2024, 8(4), 184; https://doi.org/10.3390/fractalfract8040184 - 23 Mar 2024

Cited by 2 | Viewed by 1391

Abstract

The major aim of this article is to obtain the numerical solution of a fractional mathematical model with a nonsingular kernel for thrombin receptor activation in calcium signals using two numerical schemes based on the collocation techniques. We present the computational solution of the considered fractional model using the Laguerre collocation method (LCM) and Jacobi collocation method (JCM). An operational matrix of the fractional order derivative in the Caputo sense is needed for the recommended approach. The computational scheme converts fractional differential equations (FDEs) into an algebraic set of equations using the collocation method. The technique is used more quickly and successfully than in other existing schemes. A comparison between LCM and JCM is also presented in the form of figures. We obtained very good results with a great agreement between both the schemes. Additionally, an error analysis of the suggested procedures is provided. Full article

(This article belongs to the Special Issue Feature Papers for Mathematical Physics Section)

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18 pages, 1002 KB

Open AccessArticle

A Collocation Numerical Method for Highly Oscillatory Algebraic Singular Volterra Integral Equations

by SAIRA, Wen-Xiu Ma and Guidong Liu

Fractal Fract. 2024, 8(2), 80; https://doi.org/10.3390/fractalfract8020080 - 26 Jan 2024

Viewed by 1671

Abstract

The highly oscillatory algebraic singular Volterra integral equations cannot be solved directly. A collocation numerical method is proposed to overcome the difficulty created by the highly oscillatory algebraic singular kernel. This paper is composed primarily of two methods—the piecewise constant collocation method and the piecewise linear collocation method—in which uniformly distributed nodes serve as collocation points. For the efficient computation of highly oscillatory and algebraic singular integrals, the steepest descent method as well as the Gauss–Laguerre and generalized Gauss–Laguerre quadrature rules are employed. Consequently, the resulting linear system is solved for the unknown function approximated by the Lagrange interpolation polynomial. Detailed theoretical analysis is carried out and numerical experiments showing high accuracy are also presented to confirm our analysis. Full article

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9 pages, 406 KB

Open AccessArticle

Laguerre Wavelet Approach for a Two-Dimensional Time–Space Fractional Schrödinger Equation

by Stelios Bekiros, Samaneh Soradi-Zeid, Jun Mou, Amin Yousefpour, Ernesto Zambrano-Serrano and Hadi Jahanshahi

Entropy 2022, 24(8), 1105; https://doi.org/10.3390/e24081105 - 11 Aug 2022

Cited by 1 | Viewed by 1687

Abstract

This article is devoted to the determination of numerical solutions for the two-dimensional time–spacefractional Schrödinger equation. To do this, the unknown parameters are obtained using the Laguerre wavelet approach. We discretize the problem by using this technique. Then, we solve the discretized nonlinear [...] Read more.

(This article belongs to the Special Issue Dynamical Systems, Differential Equations and Applications)

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17 pages, 465 KB

Open AccessArticle

Fibonacci Wavelet Method for the Solution of the Non-Linear Hunter–Saxton Equation

by H. M. Srivastava, Firdous A. Shah and Naied A. Nayied

Appl. Sci. 2022, 12(15), 7738; https://doi.org/10.3390/app12157738 - 1 Aug 2022

Cited by 26 | Viewed by 2322

Abstract

In this article, a novel and efficient collocation method based on Fibonacci wavelets is proposed for the numerical solution of the non-linear Hunter–Saxton equation. Firstly, the operational matrices of integration associated with the Fibonacci wavelets are constructed by following the strategy of Chen and Hsiao. The operational matrices merged with the collocation method are used to convert the given problem into a system of algebraic equations that can be solved by any classical method, such as Newton’s method. Moreover, the non-linearity arising in the Hunter–Saxton equation is handled by invoking the quasi-linearization technique. To show the efficiency and accuracy of the Fibonacci-wavelet-based numerical technique, the approximate solutions of the non-linear Hunter–Saxton equation with other numerical methods including the Haar wavelet, trigonometric B-spline, and Laguerre wavelet methods are compared. The numerical outcomes demonstrate that the proposed method yields a much more stable solution and a better approximation than the existing ones. Full article

(This article belongs to the Section Applied Physics General)

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18 pages, 654 KB

Open AccessArticle

Generalized Bessel Polynomial for Multi-Order Fractional Differential Equations

by Mohammad Izadi and Carlo Cattani

Symmetry 2020, 12(8), 1260; https://doi.org/10.3390/sym12081260 - 30 Jul 2020

Cited by 42 | Viewed by 3919

Abstract

The main goal of this paper is to define a simple but effective method for approximating solutions of multi-order fractional differential equations relying on Caputo fractional derivative and under supplementary conditions. Our basis functions are based on some original generalization of the Bessel polynomials, which satisfy many properties shared by the classical orthogonal polynomials as given by Hermit, Laguerre, and Jacobi. The main advantages of our polynomials are two-fold: All the coefficients are positive and any collocation matrix of Bessel polynomials at positive points is strictly totally positive. By expanding the unknowns in a (truncated) series of basis functions at the collocation points, the solution of governing differential equation can be easily converted into the solution of a system of algebraic equations, thus reducing the computational complexities considerably. Several practical test problems also with some symmetries are given to show the validity and utility of the proposed technique. Comparisons with available exact solutions as well as with several alternative algorithms are also carried out. The main feature of our approach is the good performance both in terms of accuracy and simplicity for obtaining an approximation to the solution of differential equations of fractional order. Full article

(This article belongs to the Special Issue Bifurcation and Chaos in Fractional-Order Systems)

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12 pages, 165 KB

Open AccessArticle

Numerical Approach of High-Order Linear Delay Difference Equations with Variable Coefficients in Terms of Laguerre Polynomials

by Burcu Gürbüz, Mustafa Gülsu and Mehmet Sezer

Math. Comput. Appl. 2011, 16(1), 267-278; https://doi.org/10.3390/mca16010267 - 1 Apr 2011

Cited by 7 | Viewed by 1793

Abstract

This paper presents a numerical method for the approximate solution of mthorder linear delay difference equations with variable coefficients under the mixed conditions in terms of Laguerre polynomials. The aim of this article is to present an efficient numerical procedure for solving mth-order linear delay difference equations with variable coefficients. Our method depends mainly on a Laguerre series expansion approach. This method transforms linear delay difference equations and the given conditions into matrix equation which corresponds to a system of linear algebraic equation. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments and performed on the computer algebraic system Maple. Full article

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