Chebyshev polynomial solutions of second-order linear partial differential equations

Keşan, Cenk

doi:10.1016/s0096-3003(01)00273-9

Cited by 19 publications

(8 citation statements)

References 6 publications

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“…Note that all the figures bellow (1)(2)(3)(4)(5) represent the relations between the projection curves of the solutions obtained from the exact solution, the finite difference method, and the proposed algorithms.…”

Section: Numerical Results and Comparisonsmentioning

confidence: 99%

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An efficient algorithm for the solution of second order PDE’s using Taylor expansion polynomials

Al-Towaiq¹,

Al-Bzoor²

2011

Journal of Discrete Mathematical Sciences and Cryptography

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Taylor expansion method for the solution of second order PDE's, is based on taking truncated Taylor polynomial about any point of the coefficients functions has been investigated by Kesan [3]. In this paper, we develop an algorithm for the solution of the PDE's of the form ( , )yy with any boundary conditions, based on taking truncated Taylor polynomials about any point of the coefficients functions to obtain the augmented systems. We present a modified sparse technique to solve the obtained systems. We verify our proposed algorithms via numerical experiments, then we compare it with Kesan's work and the finite difference method. The numerical results indicate that our algorithm is easily applicable on a wide range of PDE's and give more accurate solutions than Kesan's and the finite difference method. Also, it provides a reduction of both computational time and storage requirements without sacrificing the numerical stability.

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Section: Numerical Results and Comparisonsmentioning

confidence: 99%

“…Kesan [5], used Chebyshev polynomial to approximate the solution of second-order PDE's with two variables and variable coefficients, using the truncated Chebyshev expansions of the functions in the PDE's. Hence, the result matrix equation can be solved and the unknown Chebyshev coefficients can be found approximately.…”

Section: Introductionmentioning

confidence: 99%

An efficient algorithm for the solution of second order PDE’s using Taylor expansion polynomials

Al-Towaiq¹,

Al-Bzoor²

2011

Journal of Discrete Mathematical Sciences and Cryptography

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“…Tezer-Sezgin and Aydın (2006) Dehghan and Mirzaei (2009) proposed a technique referred as meshless local boundary integral equation method (LBIE) in order to obtain numerical solutions for coupled velocity and magnetic field equations for unsteady MHD rectangular and circular cross-sectioned duct flows with non-conducting walls. On the other hand, spectral collocation based methods have been used in pure and applied mathematics, e., g., Sezer and Kaynak (1996), Akyüz and Sezer (2003), Akyüz-Daşcıoğlu and Sezer (2005), Çelik (2005a), Çelik (2005b), Çelik and Gokmen (2005), Çelik (2006), Keşan (2003), but also in MHD duct flow problems for the coupled steady case by Çelik (2011) and in the inductionless approach for the steady case by Cuevas et al (1997). A combination of finite volume element method and spectral method is proposed in Shakeri and Dehghan (2011) for the coupled velocity and magnetic field rectangular cross-section unsteady case, focusing on building up and evaluating the method viability in terms of correctly combining the two techniques and establishing its validation respect to available analytical solutions as well as numerical ones for Hartmann numbers of 2 ( 10 ).…”

Section: Introductionmentioning

confidence: 99%

Numerical Study on the Magnetohydrodynamics of a Liquid Metal Oscillatory Flow under Inductionless Approximation

Sierra¹

2017

JAFM

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A harmonically-driven, incompressible, electrically conducting, and viscous liquid metal magnetohydrodynamic flow through a thin walled duct of rectangular cross section interacting with a uniform magnetic field traverse to its motion direction is numerically investigated. Chebyshev spectral collocation method is used to solve the Navier-Stokes equation under the inductionless approximation for the magnetic field in the gradient formulation for the electric field. Flow is considered fully developed in the direction perpendicular to the applied magnetic field and laminar in regime. Validation of numerical calculations respect to analytical calculations is established. Flow structure and key magnetohydrodynamic features regarding eventual alternating power generation application in a rectangular channel liquid metal magnetohydrodynamic generator setup are numerically inquired. Influence of pertinent parameters such as Hartmann number, oscillatory interaction parameter and wall conductance ratio on magnetohydrodynamic flow characteristics is illustrated. Particularly, it is found that in the side layer and its vicinity the emerging flow structures/patterns depend mainly on the Hartmann number and oscillatory interaction parameter ratio, while the situation for the Hartmann layer and its vicinity is less eventful. A similar feature has been discussed in the literature for the steady liquid metal flow case and served as rationale for developing the composite core-side-layer approximation to study the magnetohydrodynamics of liquid metal flows usable in direct power generation. In this study that approximation is not considered and the analysis is performed on liquid metal oscillatory (i., e., unsteady) flows usable in alternating power generation. Conversely, in terms of prospective practical applicability the formulation developed and tested with these calculations admits the implementation of a load resistance and walls conductivity optimization. That means that besides representing a numerical study on the magnetohydrodynamics of the oscillatory flow under consideration, absent in the literature for the parametric ranges reported, the formulation presently implemented can also be applicable to study the performance of an alternating liquid metal magnetohydrodynamic generator in the rectangular channel configuration.

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“…Barrett [11] determines the flow in a straight channel under a variety of wall conductivity conditions when a uniform magnetic field is imposed to the flow direction by using finite element methods. Celik [12] solve the MHD flow equations in a rectangular duct in presence of transverse external oblique magnetic field by Chebyshev collocation method [13]. AlKhawaja et.…”

Section: Introductionmentioning

confidence: 99%

Numerical Study of Liquid Metal MHD Flow through a Square Duct under the Action of Strong Transverse Magnetic Field

Sarma¹,

Hazarika²,

Deka³

2013

IJCA

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Numerical solution for steady MHD flow of liquid metal through a square duct under the action of strong transverse magnetic field has been investigated. The walls of the duct are considered to be electrically insulated as well as isothermal. The numerical solutions for velocity and temperature distributions have been obtained by finite difference method. The solutions for different values of Hartmann number and Prandtl number has been analyzed and are presented graphically. The MHD effect on velocity field and temperature field has been predicted in this investigation. KeywordsMHD flow, liquid metal flow, electrically insulated wall, square duct.

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Chebyshev polynomial solutions of second-order linear partial differential equations

Cited by 19 publications

References 6 publications

An efficient algorithm for the solution of second order PDE’s using Taylor expansion polynomials

An efficient algorithm for the solution of second order PDE’s using Taylor expansion polynomials

Numerical Study on the Magnetohydrodynamics of a Liquid Metal Oscillatory Flow under Inductionless Approximation

Numerical Study of Liquid Metal MHD Flow through a Square Duct under the Action of Strong Transverse Magnetic Field

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