scholarly journals Numerical Approximation of One- and Two-Dimensional Coupled Nonlinear Schrödinger Equation by Implementing Barycentric Lagrange Interpolation Polynomial DQM

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Varun Joshi ◽  
Mamta Kapoor ◽  
Nitin Bhardwaj ◽  
Mehedi Masud ◽  
Jehad F. Al-Amri
Keyword(s):  
Differential Equations ◽  
Lagrange Interpolation ◽  
Numerical Solutions ◽  
Differential Quadrature Method ◽  
Strong Stability ◽  
Interpolation Polynomial ◽  
Error Norm ◽  
Nonlinear Schrödinger ◽  
The Matrix ◽  
Matrix Stability

In this paper, a new numerical method named Barycentric Lagrange interpolation-based differential quadrature method is implemented to get numerical solution of 1D and 2D coupled nonlinear Schrödinger equations. In the present study, spatial discretization is done with the aid of Barycentric Lagrange interpolation basis function. After that, a reduced system of ordinary differential equations is solved using strong stability, preserving the Runge-Kutta 43 method. In order to check the accuracy of the proposed scheme, we have used the formula of L ∞ error norm. The matrix stability analysis method is implemented to test the proposed method’s stability, which confirms that the proposed scheme is unconditionally stable. The present scheme produces better results, and it is easy to implement to obtain numerical solutions of a class of partial differential equations.

scholarly journals Fuzzy Volterra Integro-Differential Equations Using General Linear Method

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 381 ◽  
Author(s):  
Zanariah Abdul Majid ◽  
Faranak Rabiei ◽  
Fatin Abd Hamid ◽  
Fudziah Ismail
Keyword(s):  
Differential Equations ◽  
Lagrange Interpolation ◽  
Linear Method ◽  
Interpolation Polynomial ◽  
General Linear ◽  
Kutta Method ◽  
Lagrange Interpolation Polynomial ◽  
Runge Kutta Method ◽  
General Linear Method ◽  
Generalized Hukuhara Differentiability

In this paper, a fuzzy general linear method of order three for solving fuzzy Volterra integro-differential equations of second kind is proposed. The general linear method is operated using the both internal stages of Runge-Kutta method and multivalues of a multisteps method. The derivation of general linear method is based on the theory of B-series and rooted trees. Here, the fuzzy general linear method using the approach of generalized Hukuhara differentiability and combination of composite Simpson’s rules together with Lagrange interpolation polynomial is constructed for numerical solution of fuzzy volterra integro-differential equations. To illustrate the performance of the method, the numerical results are compared with some existing numerical methods.

A new perspective for the numerical solutions of the cmKdV equation via modified cubic B-spline differential quadrature method

2018 ◽  
Vol 29 (06) ◽  
pp. 1850043 ◽  
Author(s):  
Ali Başhan ◽  
N. Murat Yağmurlu ◽  
Yusuf Uçar ◽  
Alaattin Esen
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Numerical Solutions ◽  
Differential Quadrature Method ◽  
Differential Quadrature ◽  
Present Approach ◽  
Test Problems ◽  
Quadrature Method ◽  
Error Norm ◽  
B Spline

In the present paper, a novel perspective fundamentally focused on the differential quadrature method using modified cubic B-spline basis functions are going to be applied for obtaining the numerical solutions of the complex modified Korteweg–de Vries (cmKdV) equation. In order to test the effectiveness and efficiency of the present approach, three test problems, that is single solitary wave, interaction of two solitary waves and interaction of three solitary waves will be handled. Furthermore, the maximum error norm [Formula: see text] will be calculated for single solitary wave solutions to measure the efficiency and the accuracy of the present approach. Meanwhile, the three lowest conservation quantities will be calculated and also used to test the efficiency of the method. In addition to these test tools, relative changes of the invariants will be calculated and presented. In the end of these processes, those newly obtained numerical results will be compared with those of some of the published papers. As a conclusion, it can be said that the present approach is an effective and efficient one for solving the cmKdV equation and can also be used for numerical solutions of other problems.

scholarly journals On the consistency of two-phase local/nonlocal piezoelectric integral model

2021 ◽  
Vol 42 (11) ◽  
pp. 1581-1598
Author(s):  
Yanming Ren ◽  
Hai Qing
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Numerical Solutions ◽  
Differential Quadrature Method ◽  
Integral Model ◽  
Governing Equations ◽  
Two Phase ◽  
Differential Model ◽  
Piezoelectric Beam ◽  
General Strain

AbstractIn this paper, we propose general strain- and stress-driven two-phase local/nonlocal piezoelectric integral models, which can distinguish the difference of nonlocal effects on the elastic and piezoelectric behaviors of nanostructures. The nonlocal piezoelectric model is transformed from integral to an equivalent differential form with four constitutive boundary conditions due to the difficulty in solving intergro-differential equations directly. The nonlocal piezoelectric integral models are used to model the static bending of the Euler-Bernoulli piezoelectric beam on the assumption that the nonlocal elastic and piezoelectric parameters are coincident with each other. The governing differential equations as well as constitutive and standard boundary conditions are deduced. It is found that purely strain- and stress-driven nonlocal piezoelectric integral models are ill-posed, because the total number of differential orders for governing equations is less than that of boundary conditions. Meanwhile, the traditional nonlocal piezoelectric differential model would lead to inconsistent bending response for Euler-Bernoulli piezoelectric beam under different boundary and loading conditions. Several nominal variables are introduced to normalize the governing equations and boundary conditions, and the general differential quadrature method (GDQM) is used to obtain the numerical solutions. The results from current models are validated against results in the literature. It is clearly established that a consistent softening and toughening effects can be obtained for static bending of the Euler-Bernoulli beam based on the general strain- and stress-driven local/nonlocal piezoelectric integral models, respectively.

scholarly journals Visualizing some numerical solutions of linear second order differential equations with the Euler method and Lagrange interpolation via GeoGebra

2021 ◽  
Vol 2090 (1) ◽  
pp. 012090
Author(s):  
Jorge Olivares Funes ◽  
Elvis Valero Kari
Keyword(s):  
Numerical Calculation ◽  
Differential Equations ◽  
Lagrange Interpolation ◽  
Numerical Solutions ◽  
Euler Method ◽  
Second Order ◽  
Teaching Material ◽  
Second Order Differential Equations

Abstract In this paper we will show the algebraic and graphic expressions, that were obtained through the Euler method and Lagrange interpolation by means of GeoGebra software for some linear second order differential equations. This teaching material was designed for the course of differential equations, and as a complement of support for the numerical calculation course for the engineering careers of the Universidad de Antofagasta.

Numerical Solutions Based on a Collocation Method Combined with Euler Polynomials for Linear Fractional Differential Equations with Delay

2020 ◽  
Vol 21 (6) ◽  
pp. 539-547
Author(s):  
Ali Konuralp ◽  
Sercan Öner
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Fractional Differential Equations ◽  
Numerical Solutions ◽  
Analytic Solutions ◽  
Euler Polynomials ◽  
The Matrix ◽  
Fractional Differential ◽  
Equations With Delay ◽  
Linear Fractional Differential Equations

AbstractIn this study, a method combined with both Euler polynomials and the collocation method is proposed for solving linear fractional differential equations with delay. The proposed method yields an approximate series solution expressed in the truncated series form in which terms are constituted of unknown coefficients that are to be determined according to Euler polynomials. The matrix method developed for the linear fractional differential equations is improved to the case of having delay terms. Furthermore, while putting the effect of conditions into the algebraic system written in the augmented form in which the coefficients of Euler polynomials are unknowns, the condition matrix scans the rows one by one. Thus, by using our program written in Mathematica there can be obtained more than one semi-analytic solutions that approach to exact solutions. Some numerical examples are given to demonstrate the efficiency of the proposed method.

scholarly journals Numerical Solutions of Fractional Differential Equations by Using Fractional Explicit Adams Method

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1675
Author(s):  
Nur Amirah Zabidi ◽  
Zanariah Abdul Majid ◽  
Adem Kilicman ◽  
Faranak Rabiei
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Lagrange Interpolation ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Multistep Method ◽  
Stability And Convergence ◽  
Third Order ◽  
Fractional Differential ◽  
Derivatives Of

Differential equations of fractional order are believed to be more challenging to compute compared to the integer-order differential equations due to its arbitrary properties. This study proposes a multistep method to solve fractional differential equations. The method is derived based on the concept of a third-order Adam–Bashforth numerical scheme by implementing Lagrange interpolation for fractional case, where the fractional derivatives are defined in the Caputo sense. Furthermore, the study includes a discussion on stability and convergence analysis of the method. Several numerical examples are also provided in order to validate the reliability and efficiency of the proposed method. The examples in this study cover solving linear and nonlinear fractional differential equations for the case of both single order as α∈(0,1) and higher order, α∈1,2, where α denotes the order of fractional derivatives of Dαy(t). The comparison in terms of accuracy between the proposed method and other existing methods demonstrate that the proposed method gives competitive performance as the existing methods.

scholarly journals Haar Wavelet Collocation Method for Solving Riccati and Fractional Riccati Differential Equations

2016 ◽  
Vol 17 ◽  
pp. 46-56 ◽  
Author(s):  
S.C. Shiralashetti ◽  
A.B. Deshi
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Numerical Solutions ◽  
Operational Matrix ◽  
Haar Wavelet ◽  
Algebraic Equations ◽  
Riccati Differential Equations ◽  
Wavelet Collocation Method ◽  
Linear Algebraic Equations ◽  
The Matrix

In this paper, numerical solutions of Riccati and fractional Riccati differential equations are obtained by the Haar wavelet collocation method. An operational matrix of integration based on the Haar wavelet is established, and the procedure for applying the matrix to solve these equations. The fundamental idea of Haar wavelet method is to convert the proposed differential equations into a group of non-linear algebraic equations. The accuracy of approximate solution can be further improved by increasing the level of resolution and an error analysis is computed. The examples are given to demonstrate the fast and flexibility of the method. The results obtained are in good agreement with the exact in comparison with existing ones and it is shown that the technique introduced here is robust, easy to apply and is not only enough accurate but also quite stable.

Numerical method for a system of integro-differential equations by Lagrange interpolation

2016 ◽  
Vol 09 (04) ◽  
pp. 1650077 ◽  
Author(s):  
Yousef Jafarzadeh ◽  
Bagher Keramati
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Lagrange Interpolation ◽  
Higher Order ◽  
Matrix Equations ◽  
Algebraic Equations ◽  
Lagrange Polynomial ◽  
Fredholm Equations ◽  
Linear Algebraic Equations ◽  
The Matrix

In this paper, we present the Lagrange polynomial solutions to system of higher-order linear integro-differential Volterra–Fredholm equations (IDVFE). This method transforms the IDVFE into the matrix equations which is converted to a system of linear algebraic equations. Some numerical results are given to illustrate the efficiency of the method.

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