Exponential Jacobi spectral method for hyperbolic partial differential equations

Youssri, Y. H.; Hafez, Ramy M.

doi:10.1007/s40096-019-00304-w

Cited by 24 publications

(15 citation statements)

References 22 publications

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“…In Table 3, we lists the maximum absolute errors at N = 16 and ν 1 = ν 2 = 1 for different values of θ . The results are more accurate than those obtained using the generalized Laguerre polynomials [40] and the exponential Jacobi functions [41]. In Figure 1, we display the log scale L ∞ −error at different values of ν 1 , ν 2 , N = M and θ = 3, ϑ = − 50.…”

Section: Applications To First‐order Hyperbolic Equationsmentioning

confidence: 80%

On Romanovski–Jacobi polynomials and their related approximation results

Abo‐Gabal

Zaky

Hafez

et al. 2020

Numerical Methods Partial

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The aim of this article is to present the essential properties of a finite class of orthogonal polynomials related to the probability density function of the F-distribution over the positive real line. We introduce some basic properties of the Romanovski-Jacobi polynomials, the Romanovski-Jacobi-Gauss type quadrature formulae and the associated interpolation, discrete transforms, spectral differentiation and integration techniques in the physical and frequency spaces, and basic approximation results for the weighted projection operator in the nonuniformly weighted Sobolev space. We discuss the relationship between such kinds of finite orthogonal polynomials and other classes of infinite orthogonal polynomials. Moreover, we derive spectral Galerkin schemes based on a Romanovski-Jacobi expansion in space and time to solve the Cauchy problem for a scalar linear hyperbolic equation in one and two space dimensions posed in the positive real line. Two numerical examples demonstrate the robustness and accuracy of the schemes. KEYWORDS differentiation matrices, finite orthogonal polynomials, Gauss-type quadrature, Romanovski-Jacobi polynomials, spectral methods 1 INTRODUCTION The different families of classical orthogonal polynomials are nowadays part of the basic mathematical machinery of numerous physical, engineering, and mathematical algorithms and methodologies [1-3]. The classical hypergeometric polynomials of Hermite, Laguerre, and Jacobi are infinite types

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Section: Applications To First‐order Hyperbolic Equationsmentioning

confidence: 80%

On Romanovski–Jacobi polynomials and their related approximation results

Abo‐Gabal

Zaky

Hafez

et al. 2020

Numerical Methods Partial

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“…We are using Jacobi polynomials as a basis function for the scheme due to its good convergence. The details of convergence of Jacobi polynomials can be found in [46][47][48][49][50]. Some recent papers on local and nonlocal boundary problems can be found in [50][51][52][53][54].…”

Section: Introductionmentioning

confidence: 99%

“…The details of convergence of Jacobi polynomials can be found in [46][47][48][49][50]. Some recent papers on local and nonlocal boundary problems can be found in [50][51][52][53][54].…”

Section: Introductionmentioning

confidence: 99%

Solving a class of local and nonlocal elliptic boundary value problems arising in heat transfer

Singh¹

2021

Heat Trans

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This paper deals with a class of Bratu's type, Troesch's, and nonlocal elliptic boundary value problems arising in the heat transfer process. Due to the strong nonlinearity and presence of parameter δ, it is very difficult to solve these problems analytically as well as numerically. By using the Jacobi spectral collocation method, these problems are solved fruitfully. We have shown the numerical as well as theoretical convergence of the suggested scheme. Numerical results are presented through figures and tables, demonstrating the accuracy of the scheme. Results are compared with some known methods to highlight its neglectable error.

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“…The shifted Gegenbauer-Gauss collocatio technique is introduced in Reference [ 19 ], for functional-differential equations. The exponential Jacobi spectral and Jacobi collocation approaches are developed in References [ 20 , 21 ].…”

Section: Introductionmentioning

confidence: 99%

Machine Learning for Modeling the Singular Multi-Pantograph Equations

Mosavi

Shokri

Mansor

et al. 2020

Entropy

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In this study, a new approach to basis of intelligent systems and machine learning algorithms is introduced for solving singular multi-pantograph differential equations (SMDEs). For the first time, a type-2 fuzzy logic based approach is formulated to find an approximated solution. The rules of the suggested type-2 fuzzy logic system (T2-FLS) are optimized by the square root cubature Kalman filter (SCKF) such that the proposed fineness function to be minimized. Furthermore, the stability and boundedness of the estimation error is proved by novel approach on basis of Lyapunov theorem. The accuracy and robustness of the suggested algorithm is verified by several statistical examinations. It is shown that the suggested method results in an accurate solution with rapid convergence and a lower computational cost.

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Exponential Jacobi spectral method for hyperbolic partial differential equations

Cited by 24 publications

References 22 publications

On Romanovski–Jacobi polynomials and their related approximation results

On Romanovski–Jacobi polynomials and their related approximation results

Solving a class of local and nonlocal elliptic boundary value problems arising in heat transfer

Machine Learning for Modeling the Singular Multi-Pantograph Equations

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