Jacobi rational   approximation and  spectral  method for differential equations  of degenerate type

Wang, Zhongqing; Guo, Bing

doi:10.1090/s0025-5718-07-02074-1

Cited by 26 publications

(13 citation statements)

References 25 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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“…[20]. However, in this paper, we use various orthogonal systems induced by generalized Jacobi functions given in [9,10].…”

Section: Remark 34mentioning

confidence: 99%

“…It is noted that many partial differential equations arising in financial mathematics can be reformed to the above equation, such as Black-Scholes, Dothan and Black-Derman-Toy models, cf. [16,20]. In the next two subsections, we shall design proper spectral schemes for (4.1) based on the boundary condition at the infinity.…”

Section: A Degenerated Model Problemmentioning

confidence: 99%

“…But the method in [16] is only available for solutions decaying to zero as x → ∞, while the method in [20] can be used only for solutions growing up as x → ∞. Whereas, our new method are applicable to a large class of problems with various asymptotic behaviors at the infinity.…”

Section: Remark 43mentioning

confidence: 99%

“…Thus, their weight functions are fixed, which might not be most appropriate in some cases. Wang and Guo [20] provided the Jacobi rational orthogonal approximation on the half line. Since the related orthogonal system is induced by the standard Jacobi polynomials, its weight function is still fixed in certain sense.…”

Section: Introductionmentioning

confidence: 99%

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Generalized Jacobi rational spectral method on the half line

Guo

2011

Adv Comput Math

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We introduce an orthogonal system on the half line, induced by generalized Jacobi functions. Some basic results on the generalized Jacobi rational approximation are established, which play important roles in the related spectral method. As an example of applications, the rational spectral method is proposed for partial differential equations of degenerate type. Its convergence is proved. Numerical results demonstrate its efficiency.Keywords Generalized Jacobi rational approximation · Spectral method on the half line · Applications Mathematics Subject Classification (2010) 65M70 · 41A30 · 35K65 Communicated by Zhongying Chen.

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Sobolev orthogonal Legendre rational spectral methods for problems on the half line

Shan

Wang

2019

Math Methods in App Sciences

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Modified Legendre rational spectral methods for solving second-order differential equations on the half line are proposed. Some Sobolev orthogonal Legendre rational basis functions are constructed, which lead to the diagonalization of discrete systems. Accordingly, both the exact solutions and the approximate solutions can be represented as infinite and truncated Fourier series. Numerical results demonstrate the effectiveness and the spectral accuracy of this approach. KEYWORDS elliptic boundary value problems, modified Legendre rational spectral methods, numerical results, Sobolev orthogonal functions MSC CLASSIFICATION 65N35; 41A20; 32C45; 35J25; 35K05 INTRODUCTIONSpectral method is an extremely effective tool for solving various problems in science and engineering numerically. For unbounded domains, we usually employ the Hermite or Laguerre polynomials/functions as the basis functions to approximate the potential problems; see, eg, previous studies. 1-11 However, since the Laguerre and Hermite Gauss points are too concentrated near zero, the approximation results are usually not ideal for algebraic decay functions, especially where the points are far away from zero. Alternately, we may use rational functions to approximate the problems on unbounded domains. Christov, 12 Boyd, 13,14 and Guo et al [15][16][17][18] developed various rational spectral methods on infinite intervals. Clearly, the distribution of the Gauss points for the rational functions is more reasonable, and the approximation results are also ideal especially for algebraic decay solutions. Nevertheless, the utilization of rational spectral methods usually lead to a full or sparse algebraic system as mentioned in Section 3 of this paper, the condition numbers increase as (N 2 ) for the second order problems.As is well known, the Fourier functions are the eigenfunctions of the Laplace operator, whose kth derivatives are orthogonal with respect to each other. This property results in the diagonalization of the resulting algebraic systems of the corresponding Fourier spectral methods for periodic problems; cf previous studies. 2,3,10 For nonperiodic problems, the usual spectral methods only lead to a highly sparse or full algebraic system. It is really meaningful and worth looking forward to find out a class of basis functions such that the resulting algebraic systems of the spectral methods for nonperiodic problems are still diagonal; cf previous studies. 19 Recently, Liu et al 20,21 constructed the Fourier-like Sobolev orthogonal basis functions based on generalized Laguerre functions and applied them to the second-and fourth-order elliptic equations on the half line. Motivated by previous studies, 19-21 the main purpose of this paper is to construct the Fourier-like Legendre rational basis functions on the half line, such that they are mutually orthogonal with respect to certain Sobolev inner product. On this basis, we further propose the diagonalized Legendre rational spectral methods for the second-order elliptic boundary value problems with D...

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Jacobi rational approximation and spectral method for differential equations of degenerate type

Cited by 26 publications

References 25 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

Generalized Jacobi rational spectral method on the half line

Sobolev orthogonal Legendre rational spectral methods for problems on the half line

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