Laguerre Polynomial Solutions of a Class of Initial and Boundary Value Problems Arising in Science and Engineering Fields

Gürbüz, Burcu; Sezer, Mehmet

doi:10.12693/aphyspola.130.194

Cited by 38 publications

(25 citation statements)

References 11 publications

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“…Partial differential equations of fractional order, as generalizations of classical integer order partial differential equations, are increasingly used in model problems of fluid flow, physical and biological processes, control, engineering and systems [1][2][3][4][5]. Most fractional differential equations do not have exact analytical solutions, therefore, approximation and numerical techniques are used extensively for solving these equations [6].…”

Section: Introductionmentioning

confidence: 99%

Numerical Solution of Fractional Black-Scholes Equation by Using the Multivariate Padé Approximation

Özdemir

Yavuz

2017

Acta Phys. Pol. A

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In this study, a new application of multivariate Padé approximation method has been used for solving European vanilla call option pricing problem. Padé polynomials have occurred for the fractional Black-Scholes equation, according to the relations of "smaller than", or "greater than", between stock price and exercise price of the option. Using these polynomials, we have applied the multivariate Padé approximation method to our fractional equation and we have calculated numerical solutions of fractional Black-Scholes equation for both of two situations. The obtained results show that the multivariate Padé approximation is a very quick and accurate method for fractional Black-Scholes equation. The fractional derivative is understood in the Caputo sense.

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Section: Introductionmentioning

confidence: 99%

Numerical Solution of Fractional Black-Scholes Equation by Using the Multivariate Padé Approximation

Özdemir

Yavuz

2017

Acta Phys. Pol. A

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“…. , N ) are unknown Laguerre polynomial coefficients, and N is any integer such that N ≥ 2 [17,18]. * corresponding author; e-mail: burcu_grbz@yahoo.com 2.…”

Section: Introductionmentioning

confidence: 99%

A New Computational Method Based on Laguerre Polynomials for Solving Certain Nonlinear Partial Integro Differential Equations

Gürbüz

Sezer

2017

Acta Phys. Pol. A

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In this study, we consider some nonlinear partial integro-differential equations. Most of these equations are used as mathematical models in many problems of physics, biology, chemistry, engineering, and in other areas. Our main purpose is to propose a new numerical method based on the Laguerre and Taylor polynomials, called matrix collocation method, for the numerical solution of the mentioned nonlinear equations under the initial or boundary conditions. To show the effectiveness of this approach, some examples along with error estimations are illustrated by tables and figures.

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“…This study is devoted to the application of the Laguerre collocation method to the numerical solution of the 1‐dimensional parabolic convection‐diffusion equation

\frac{\partial u}{\partial t} = \frac{\partial^{2} u}{\partial x^{2}} + A false(x false) \frac{\partial u}{\partial x} + B false(x false) u + f false(x, t false), 0 \leq x \leq l, 0 \leq t \leq T,

with the initial conditions

u false(x, 0 false) = g false(x false), 0 \leq x < \infty,

and the boundary conditions

u false(0, t false) = h false(t false), u false(l, t false) = K false(t false), 0 \leq t \leq l \leq T < \infty,

where f ( x , t ), A ( x ), B ( x ), g ( x ), and h ( t ) are functions defined in [0, l ]×[0, T ]; l and T are appropriate constants. In this study, we develop the Laguerre collocation method given in Gürbüz and Sezer and use to obtain the approximate solution of Equation in the truncated Laguerre series form

u false(x, t false) = true \sum_{r = 0}^{N} true \sum_{s = 0}^{N} a_{r, s} L_{r, s} false(x, t false); L_{r, s} false(x, t false) = L_{r} false(x false) L_{s} false(t false),

where a r , s r , s =0,…, N are the unknown Laguerre coefficients and L n ( x ), n =0,1,2,…, N are the Laguerre polynomials defined by

L_{n} false(x false) = true \sum_{k = 0}^{n} false(- 1 false)

…”

Section: Introductionmentioning

confidence: 99%

Modified Laguerre collocation method for solving 1‐dimensional parabolic convection‐diffusion problems

Gürbüz

Sezer

2017

Math Methods in App Sciences

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In this study, we propose a modified Laguerre collocation method based on operational matrix technique to solve 1-dimensional parabolic convection-diffusion problems arising in applied sciences. The method transforms the equation and mixed conditions of problem into a matrix equation with unknown Laguerre coefficients by means of collocation points and operational matrices. The solution of this matrix equation yields the Laguerre coefficients of the solution function. Thereby, the approximate solution is obtained in the truncated Laguerre series form. Also, to illustrate the usefulness and applicability of the method, we apply it to a test problem together with residual error estimation and compare the results with existing ones. Besides, the algorithm of the present method is given to represent the calculation of approximate solution.

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Laguerre Polynomial Solutions of a Class of Initial and Boundary Value Problems Arising in Science and Engineering Fields

Cited by 38 publications

References 11 publications

Numerical Solution of Fractional Black-Scholes Equation by Using the Multivariate Padé Approximation

Numerical Solution of Fractional Black-Scholes Equation by Using the Multivariate Padé Approximation

A New Computational Method Based on Laguerre Polynomials for Solving Certain Nonlinear Partial Integro Differential Equations

Modified Laguerre collocation method for solving 1‐dimensional parabolic convection‐diffusion problems

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