The spectral collocation method with three different bases for solving a nonlinear partial differential equation arising in modeling of nonlinear waves

Dehghan, Mehdi; Fakhar-Izadi, Farhad

doi:10.1016/j.mcm.2011.01.011

Cited by 58 publications

(24 citation statements)

References 32 publications

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“…Many researchers have used those methods for the numerical solution of nonlinear PDEs [25], fractional ODEs [26], high-order boundary value problems [27], systems of Volterra integral equations [28], optimal control problems governed by Volterra integral equations [29], Quasi Bang-Bang optimal control problems [30], and ODEs of degenerate types [31]. In relation to many other methods, spectral methods give highly accurate results.…”

Section: The Chebyshev Pseudo-spectral Methodsmentioning

confidence: 99%

A parameterized multi-step Newton method for solving systems of nonlinear equations

Ahmad

Tohidi²,

Carrasco³

2015

Numer Algor

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We construct a novel multi-step iterative method for solving systems of nonlinear equations by introducing a parameter θ to generalize the multi-step Newton method while keeping its order of convergence and computational cost. By an appropriate selection of θ, the new method can both have faster convergence and have larger radius of convergence. The new iterative method only requires one Jacobian inversion per iteration, and, therefore, can be efficiently implemented using Krylov subspace methods. The new method can be used to solve nonlinear systems of partial differential equations, such as complex generalized Zakharov systems of partial differential equations, by transforming them into systems of nonlinear equations by discretizing approaches in both spacial and temporal dimensions such as, for instance, the Chebyshev pseudospectral discretizing method. Quite extensive tests show that the new method can have significantly faster convergence and significantly larger radii of convergence than the multi-step Newton method.Keywords: Multi-step iterative methods; Multi-step Newton method; systems of nonlinear equations; partial differential equations; discretization methods for partial differential equations.

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Section: The Chebyshev Pseudo-spectral Methodsmentioning

confidence: 99%

A parameterized multi-step Newton method for solving systems of nonlinear equations

Ahmad

Tohidi²,

Carrasco³

2015

Numer Algor

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“…Here, we explicitly assume that the numerical methods used for approximating Equation (24) lead to a representation as in Equation (25). The higher order Fréchet derivatives of Equation (25) can be computed as:…”

Section: Application To Systems Of Nonlinear Equations Associated Witmentioning

confidence: 99%

Constructing Frozen Jacobian Iterative Methods for Solving Systems of Nonlinear Equations, Associated with ODEs and PDEs Using the Homotopy Method

et al. 2016

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Abstract:A homotopy method is presented for the construction of frozen Jacobian iterative methods. The frozen Jacobian iterative methods are attractive because the inversion of the Jacobian is performed in terms of LUfactorization only once, for a single instance of the iterative method. We embedded parameters in the iterative methods with the help of the homotopy method: the values of the parameters are determined in such a way that a better convergence rate is achieved. The proposed homotopy technique is general and has the ability to construct different families of iterative methods, for solving weakly nonlinear systems of equations. Further iterative methods are also proposed for solving general systems of nonlinear equations.

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“…Using Chebyshev pseudo-spectral collocation method [4,8,15], we performed the following discretization of (4.3) 4) where I denotes the identity matrix, and ⊗ is a Kronecker product and T ·· is the discretization of second order derivative. In Tables 7 and 8, we show the error in the numerical solution of the problem (4.3) for different nonlinear terms against the various grid sizes.…”

Section: -D Nonlinear Poisson Problemmentioning

confidence: 99%

Frozen jacobian iterative method for solving systems of nonlinear equations application to nonlinear IVPs and BVPs

Ullah¹,

Ahmad²,

Alzahrani³

et al. 2016

J. Nonlinear Sci. Appl.

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Frozen Jacobian iterative methods are of practical interest to solve the system of nonlinear equations. A frozen Jacobian multi-step iterative method is presented. We divide the multi-step iterative method into two parts namely base method and multi-step part. The convergence order of the constructed frozen Jacobian iterative method is three, and we design the base method in a way that we can maximize the convergence order in the multi-step part. In the multi-step part, we utilize a single evaluation of the function, solve four systems of lower and upper triangular systems and a second frozen Jacobian. The attained convergence order per multi-step is four. Hence, the general formula for the convergence order is 3 + 4(m − 2) for m ≥ 2 and m is the number of multi-steps. In a single instance of the iterative method, we employ only single inversion of the Jacobian in the form of LU factors that makes the method computationally cheaper because the LU factors are used to solve four system of lower and upper triangular systems repeatedly. The claimed convergence order is verified by computing the computational order of convergence for a system of nonlinear equations. The efficiency and validity of the proposed iterative method are narrated by solving many nonlinear initial and boundary value problems. c 2016 All rights reserved.Keywords: Frozen Jacobian iterative methods, multi-step iterative methods, systems of nonlinear equations, nonlinear initial value problems, nonlinear boundary value problems. 2010 MSC: 65H10, 65N22.

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The spectral collocation method with three different bases for solving a nonlinear partial differential equation arising in modeling of nonlinear waves

Cited by 58 publications

References 32 publications

A parameterized multi-step Newton method for solving systems of nonlinear equations

A parameterized multi-step Newton method for solving systems of nonlinear equations

Constructing Frozen Jacobian Iterative Methods for Solving Systems of Nonlinear Equations, Associated with ODEs and PDEs Using the Homotopy Method

Frozen jacobian iterative method for solving systems of nonlinear equations application to nonlinear IVPs and BVPs

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