Efficient Legendre pseudospectral method for solving integral and integro-differential equations

Numerical treatments for the generalized Burger's—Huxley GBH equation are presented. The treatments are based on cardinal Chebyshev and Legendre basis functions with Galerkin method. Gauss quadrature formula and El-gendi method are used to convert the problem into a system of ordinary differential equations. The numerical results are compared with the literatures to show efficiency of the proposed methods.

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“…Mathematical Problems in Engineering By using (20), (22), (23), (24), and (25), we can write the discrete weak form in the following form:…”

Section: Gauss Chebyshev Galerkin (Gcg) Methodsmentioning

confidence: 99%

“…Analogous to the previous section we consider the Legendre cardinal function based on Legendre Gauss-Lobatto (LGL) nodes. El-gendi approximation will be used with a linear combination of the Legendre cardinal function as follows [20]:…”

Section: El-gendi Legendre Galerkin (Elg) Methodsmentioning

confidence: 99%

Development of Galerkin Method for Solving the Generalized Burger's-Huxley Equation

El-Kady

El-Sayed

Fathy

2013

Mathematical Problems in Engineering

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Section: El-gendi Legendre Galerkin (Elg) Methodsmentioning

confidence: 99%

Chebyshev and Legendre via Galerkin method for solving KdV Burgers' equation

Fathy¹,

El-Kady²,

El-Sayed³

2013

ijcms

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In this paper, three numerical solutions for the Kortewegde Vries Burgers' (KdVB) equation are presented. Two of these methods are based on cardinal Chebyshev basis function with Galerkin method. Gauss-quadrature formula and El-gendi method are used to convert the problem into system of ordinary differential equations. In the third proposed method, the cardinal Legendre basis function with Galerkin method is used. In this case, the approximations are based on El-gendi method. The numerical results obtained by these ways have been compared with other solutions by Darvishi's preconditioning to the same problem to show the efficiency of the proposed methods.

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“…Table I shows numerical errors of (8)-(10) in predicting the quadrature formulas for the Lagrange interpolation defined using the LGL points and the CGL points. In these cases, all quadrature formulas to get I j k , D j k , w k j =N,k=N j =0,k=0 are available for the PS methods [1], [11], [12]. The results show the weights and the integration matrix by using (8) and (10) can be computed in machine precision regardless of the number of nodes.…”

Section: Extension Of G-spin Formula To Tension Spline Interpolationmentioning

confidence: 99%

Efficient ST Techniques for Nonlinear Optimal Control Analyses Using a Pseudospectral Framework

Kim

Sung

2015

IEEE Trans. Contr. Syst. Technol.

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In this brief, numerical techniques for an efficient implementation of the spline transcription (ST) method by using the pseudospectral framework are proposed. It intends to develop a new approach to the ST method, analyses with which can be carried out with nearly the same level of computational burden without resorting to a specific type of collocation points as in the pseudospectral method. Therefore, the piecewise continuous polynomial interpolation is converted into a global spline interpolation, which has the same forms as the Lagrange interpolation and is used to derive the formulas for the partial integration and the differentiation. Using these formulas, the nonlinear optimal control problem can be transcribed in a similar manner as in the pseudospectral method and analyzed with numerical techniques and mathematical theories established well for the pseudospectral method. Also, this brief shows the Lagrange and tension spline interpolation can be used for the proposed methods. Numerical accuracy and efficiency of the proposed method are accessed through its applications to the solution of two nonlinear optimal control problems on uniform nodes. The predicted results of states, controls, and costates are well correlated with the optimal solutions. Finally, this brief recommends the use of advanced grid topologies to enhance both accuracy and convergence by using the flexibility in node selections with the proposed methods.Index Terms-Collocation, Lagrange interpolation, nonlinear optimal control, pseudospectral (PS) method, spline transcription (ST).

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Efficient Legendre pseudospectral method for solving integral and integro-differential equations

Cited by 12 publications

References 10 publications

Development of Galerkin Method for Solving the Generalized Burger's-Huxley Equation

Development of Galerkin Method for Solving the Generalized Burger's-Huxley Equation

Chebyshev and Legendre via Galerkin method for solving KdV Burgers' equation

Efficient ST Techniques for Nonlinear Optimal Control Analyses Using a Pseudospectral Framework

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