scholarly journals Bernoulli Collocation Method for Solving Linear Multidimensional Diffusion and Wave Equations with Dirichlet Boundary Conditions

2017 ◽  
Vol 2017 ◽  
pp. 1-15 ◽  
Author(s):  
Bashar Zogheib ◽  
Emran Tohidi ◽  
Stanford Shateyi
Keyword(s):  
Boundary Conditions ◽  
Wave Equations ◽  
Krylov Subspace ◽  
Numerical Solutions ◽  
Dirichlet Boundary Conditions ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Collocation Points ◽  
Linear Algebraic Equations ◽  
Initial And Boundary Conditions

A numerical approach is proposed for solving multidimensional parabolic diffusion and hyperbolic wave equations subject to the appropriate initial and boundary conditions. The considered numerical solutions of the these equations are considered as linear combinations of the shifted Bernoulli polynomials with unknown coefficients. By collocating the main equations together with the initial and boundary conditions at some special points (i.e., CGL collocation points), equations will be transformed into the associated systems of linear algebraic equations which can be solved by robust Krylov subspace iterative methods such as GMRES. Operational matrices of differentiation are implemented for speeding up the operations. In both of the one-dimensional and two-dimensional diffusion and wave equations, the geometrical distributions of the collocation points are depicted for clarity of presentation. Several numerical examples are provided to show the efficiency and spectral (exponential) accuracy of the proposed method.

scholarly journals Genocchi Wavelet-like Operational Matrix and its Application for Solving Non-linear Fractional Differential Equations

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 463-472 ◽  
Author(s):  
Abdulnasir Isah ◽  
Chang Phang
Keyword(s):  
Differential Equations ◽  
Numerical Solutions ◽  
Operational Matrix ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Non Linear ◽  
Fractional Differential ◽  
First Time ◽  
Linear Fractional Differential Equations

AbstractIn this work, we propose a new operational method based on a Genocchi wavelet-like basis to obtain the numerical solutions of non-linear fractional order differential equations (NFDEs). To the best of our knowledge this is the first time a Genocchi wavelet-like basis is presented. The Genocchi wavelet-like operational matrix of a fractional derivative is derived through waveletpolynomial transformation. These operational matrices are used together with the collocation method to turn the NFDEs into a system of non-linear algebraic equations. Error estimates are shown and some illustrative examples are given in order to demonstrate the accuracy and simplicity of the proposed technique.

scholarly journals Approximate Solutions of Differential Equations by Using the Bernstein Polynomials

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Y. Ordokhani ◽  
S. Davaei far
Keyword(s):  
Differential Equations ◽  
Bernstein Polynomial ◽  
Numerical Solutions ◽  
Bernstein Polynomials ◽  
Approximate Solutions ◽  
High Accuracy ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Bernstein Polynomial Basis

A numerical method for solving differential equations by approximating the solution in the Bernstein polynomial basis is proposed. At first, we demonstrate the relation between the Bernstein and Legendre polynomials. By using this relation, we derive the operational matrices of integration and product of the Bernstein polynomials. Then, we employ them for solving differential equations. The method converts the differential equation to a system of linear algebraic equations. Finally some examples and their numerical solutions are given; comparing the results with the numerical results obtained from the other methods, we show the high accuracy and efficiency of the proposed method.

scholarly journals Numerical solutions of a class of nonlinear ordinary differential equations in Hermite series

2019 ◽  
Vol 23 (Suppl. 1) ◽  
pp. 339-351
Author(s):  
Coskun Guler ◽  
Saba Kaya ◽  
Mehmet Sezer
Keyword(s):  
Hermite Polynomial ◽  
Numerical Solutions ◽  
Nonlinear Ordinary Differential Equations ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Collocation Points ◽  
Linear Algebraic Equations ◽  
Non Linear ◽  
Existing Result ◽  
Hermite Series

The purpose of this paper is to present a Hermite polynomial approach for solving a high-order ODE with non-linear terms under mixed conditions. The method we used is a matrix method based on collocation points together with truncated Hermite series and reduces the solution of equation to solution of a matrix equation which corresponds to a system of non-linear algebraic equations with unknown Hermite coefficients. In addition, to illustrate the validity and applicability of the method, some numerical examples together with residual error analysis are performed and the obtained results are compared with the existing result in literature.

scholarly journals Two matrix methods for solution of nonlinear and linear Lane-Emden type equations with mixed condition by operational matrix

2015 ◽  
Vol 4 (3) ◽  
pp. 420 ◽  
Author(s):  
Behrooz Basirat ◽  
Mohammad Amin Shahdadi
Keyword(s):  
Bernstein Polynomials ◽  
Operational Matrix ◽  
Numerical Procedure ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Mixed Condition ◽  
Matrix Methods ◽  
Numerical Examples ◽  
Linear Algebraic Equations ◽  
The Given

<p>The aim of this article is to present an efficient numerical procedure for solving Lane-Emden type equations. We present two practical matrix method for solving Lane-Emden type equations with mixed conditions by Bernstein polynomials operational matrices (BPOMs) on interval [<em>a; b</em>]. This methods transforms Lane-Emden type equations and the given conditions into matrix equation which corresponds to a system of linear algebraic equations. We also give some numerical examples to demonstrate the efficiency and validity of the operational matrices for solving Lane-Emden type equations (LEEs).</p>

scholarly journals A Pseudospectral Algorithm for Solving Multipantograph Delay Systems on a Semi-Infinite Interval Using Legendre Rational Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
E. H. Doha ◽  
D. Baleanu ◽  
A. H. Bhrawy ◽  
R. M. Hafez
Keyword(s):  
Numerical Solutions ◽  
Rational Functions ◽  
Absolute Error ◽  
Delay Systems ◽  
Infinite Interval ◽  
Algebraic Equations ◽  
Collocation Points ◽  
Nonlinear Algebraic Equations ◽  
Linear And Nonlinear ◽  
Pseudospectral Scheme

A new Legendre rational pseudospectral scheme is proposed and developed for solving numerically systems of linear and nonlinear multipantograph equations on a semi-infinite interval. A Legendre rational collocation method based on Legendre rational-Gauss quadrature points is utilized to reduce the solution of such systems to systems of linear and nonlinear algebraic equations. In addition, accurate approximations are achieved by selecting few Legendre rational-Gauss collocation points. The numerical results obtained by this method have been compared with various exact solutions in order to demonstrate the accuracy and efficiency of the proposed method. Indeed, for relatively limited nodes used, the absolute error in our numerical solutions is sufficiently small.

Analysis of Dynamic Systems With Periodically Varying Parameters via Chebyshev Polynomials

1991 ◽  
Author(s):  
S. C. Sinha ◽  
Der-Ho Wu ◽  
Vikas Juneja ◽  
Paul Joseph
Keyword(s):  
Dynamic Systems ◽  
Chebyshev Polynomials ◽  
Numerical Solutions ◽  
Algebraic Equations ◽  
Suggested Approach ◽  
Varying Parameters ◽  
Approximate Analytical Solutions ◽  
Linear Algebraic Equations ◽  
Mathieu’S Equation ◽  
Runge Kutta

Abstract In this paper a general method for the analysis of multidimensional second-order dynamic systems with periodically varying parameters is presented. The state vector and the periodic matrices appearing in the equations are expanded in Chebyshev polynomials over the principal period and the original differential problem is reduced to a set of linear algebraic equations. The technique is suitable for constructing either numerical or approximate analytical solutions. As an illustrative example, approximate analytical expressions for the Floquet characteristic exponents of Mathieu’s equation are obtained. Stability charts are drawn to compare the results the proposed method with those obtained by Runge-Kutta and perturbation methods. Numerical solutions for the flap-lag motion of a three blade helicopter rotor are constructed in the next example. The numerical accuracy and efficiency of the proposed technique is compared with standard numerical codes based on Runge-Kutta, Adams-Moulton and Gear algorithms. The results obtained in the both examples indicate that the suggested approach extremely accurate and is by far the most efficient one.

scholarly journals Using Fractional Bernoulli Wavelets for Solving Fractional Diffusion Wave Equations with Initial and Boundary Conditions

2021 ◽  
Vol 5 (4) ◽  
pp. 212
Author(s):  
Monireh Nosrati Sahlan ◽  
Hojjat Afshari ◽  
Jehad Alzabut ◽  
Ghada Alobaidi
Keyword(s):  
Fractional Order ◽  
Fractional Derivatives ◽  
Wave Equations ◽  
Bernoulli Polynomials ◽  
Fractional Diffusion ◽  
Computational Time ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Diffusion Wave ◽  
Klein Gordon

In this paper, fractional-order Bernoulli wavelets based on the Bernoulli polynomials are constructed and applied to evaluate the numerical solution of the general form of Caputo fractional order diffusion wave equations. The operational matrices of ordinary and fractional derivatives for Bernoulli wavelets are set via fractional Riemann–Liouville integral operator. Then, these wavelets and their operational matrices are utilized to reduce the nonlinear fractional problem to a set of algebraic equations. For solving the obtained system of equations, Galerkin and collocation spectral methods are employed. To demonstrate the validity and applicability of the presented method, we offer five significant examples, including generalized Cattaneo diffusion wave and Klein–Gordon equations. The implementation of algorithms exposes high accuracy of the presented numerical method. The advantage of having compact support and orthogonality of these family of wavelets trigger having sparse operational matrices, which reduces the computational time and CPU requirements.

A Numerical Algorithm for Solving Stiff ODEs and DAEs in Multibody Mechanics

1999 ◽  
Author(s):  
Hajrudin Pasic
Keyword(s):  
Algebraic System ◽  
Numerical Solutions ◽  
Solution Procedure ◽  
Differential Algebraic Equations ◽  
Good Precision ◽  
Algebraic Equations ◽  
Constant Matrix ◽  
Fully Implicit ◽  
Collocation Points ◽  
Collocation Technique

Abstract Presented is an algorithm suitable for numerical solutions of multibody mechanics problems. When s-stage fully implicit Runge-Kutta (RK) method is used to solve these problems described by a system of n ordinary differential equations (ODE), solution of the resulting algebraic system requires 2s3 n3 / 3 operations. In this paper we present an efficient algorithm, whose formulation differs from the traditional RK method. The procedure for uncoupling the algebraic system into a block-diagonal matrix with s blocks of size n is derived for any s. In terms of number of multiplications, the algorithm is about s2 / 2 times faster than the original, nondiagonalized system, as well as s2 times in terms of number of additions/multiplications. With s = 3 the method has the same precision and stability property as the well-known RADAU5 algorithm. However, our method is applicable with any s and not only to the explicit ODEs My′ = f(x, y), where M = constant matrix, but also to the general implicit ODEs of the form f (x, y, y′) = 0. In the solution procedure y is assumed to have a form of the algebraic polynomial whose coefficients are found by using the collocation technique. A proper choice of locations of collocation points guarantees good precision/stability properties. If constructed such as to be L-stable, the method may be used for solving differential-algebraic equations (DAEs). The application is illustrated by a constrained planar manipulator problem.

scholarly journals A Collocation Approach for Solving Time-Fractional Stochastic Heat Equation Driven by an Additive Noise

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 904 ◽  
Author(s):  
Afshin Babaei ◽  
Hossein Jafari ◽  
S. Banihashemi
Keyword(s):  
Order Of Convergence ◽  
Additive Noise ◽  
Numerical Solutions ◽  
Numerical Test ◽  
Test Problems ◽  
Heat Equations ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Stochastic Heat Equations ◽  
Collocation Approach

A spectral collocation approach is constructed to solve a class of time-fractional stochastic heat equations (TFSHEs) driven by Brownian motion. Stochastic differential equations with additive noise have an important role in explaining some symmetry phenomena such as symmetry breaking in molecular vibrations. Finding the exact solution of such equations is difficult in many cases. Thus, a collocation method based on sixth-kind Chebyshev polynomials (SKCPs) is introduced to assess their numerical solutions. This collocation approach reduces the considered problem to a system of linear algebraic equations. The convergence and error analysis of the suggested scheme are investigated. In the end, numerical results and the order of convergence are evaluated for some numerical test problems to illustrate the efficiency and robustness of the presented method.

scholarly journals A Numerical Method for Time-Fractional Reaction-Diffusion and Integro Reaction-Diffusion Equation Based on Quasi-Wavelet

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Sachin Kumar ◽  
Jinde Cao ◽  
Xiaodi Li
Keyword(s):  
Diffusion Equation ◽  
Boundary Conditions ◽  
Numerical Method ◽  
Numerical Solutions ◽  
Exact Analytical Solution ◽  
Research Work ◽  
Reaction Diffusion ◽  
Dirichlet Boundary Conditions ◽  
Reaction Diffusion Equation ◽  
Fractional Reaction

In this research work, we focused on finding the numerical solution of time-fractional reaction-diffusion and another class of integro-differential equation known as the integro reaction-diffusion equation. For this, we developed a numerical scheme with the help of quasi-wavelets. The fractional term in the time direction is approximated by using the Crank–Nicolson scheme. The spatial term and the integral term present in integro reaction-diffusion are discretized and approximated with the help of quasi-wavelets. We study this model with Dirichlet boundary conditions. The discretization of these initial and boundary conditions is done with a different approach by the quasi-wavelet-based numerical method. The validity of this proposed method is tested by taking some numerical examples having an exact analytical solution. The accuracy of this method can be seen by error tables which we have drawn between the exact solution and the approximate solution. The effectiveness and validity can be seen by the graphs of the exact and numerical solutions. We conclude that this method has the desired accuracy and has a distinctive local property.

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