An efficient computational technique based on cubic trigonometric B-splines for time fractional Burgers' equation

Yaseen, Muhammad; Abbas, Muhammad

doi:10.1080/00207160.2019.1612053

Cited by 35 publications

(30 citation statements)

References 38 publications

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“…Various numerical methods based on spline functions have also been employed by researchers in pursue of reliable solutions for fractional-order differential equations [29][30][31]. The B-spline functions provide decent approximations in contrast with rest of numerical schemes due to the nominal, compact support and C 2 continuity [32].…”

Section: Introductionmentioning

confidence: 99%

A numerical algorithm based on modified extended B-spline functions for solving time-fractional diffusion wave equation involving reaction and damping terms

et al. 2019

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In this study, we have proposed an efficient numerical algorithm based on third degree modified extended B-spline (EBS) functions for solving time-fractional diffusion wave equation with reaction and damping terms. The Caputo time-fractional derivative has been approximated by means of usual finite difference scheme and the modified EBS functions are used for spatial discretization. The stability analysis and derivation of theoretical convergence validates the authenticity and effectiveness of the proposed algorithm. The numerical experiments show that the computational outcomes are in line with the theoretical expectations. Moreover, the numerical results are proved to be better than other methods on the topic.

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Section: Introductionmentioning

confidence: 99%

A numerical algorithm based on modified extended B-spline functions for solving time-fractional diffusion wave equation involving reaction and damping terms

et al. 2019

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“…where d i (t) are unknown functions to be determined and TB 3 i (x) 20,30,31 are twice differentiable cubic trigonometric basis functions given by…”

Section: Derivation Of the Schemementioning

confidence: 99%

“…The solution domain a ≤ x ≤ b is uniformly partitioned by knots x j into M subintervals [ x j , x j + 1 ] of equal length h , j = 0,1,2, … , M − 1, where a = x 0 < x 1 < … < x n − 1 < x M = b . Our scheme for solving () requires approximate solution U ( x , t ) to the exact solution u ( x , t ) in the following form: 25‐29

U false(x, t false) = true \sum_{i = - 1}^{N - 1} d_{i} false(t false) T B_{i}^{3} false(x false),

where d i ( t ) are unknown functions to be determined and

T B_{i}^{3} false(x false)

20,30,31 are twice differentiable cubic trigonometric basis functions given by

T B_{i}^{3} false(x false) = \frac{1}{w} ({, \begin{matrix} left left \\ array y^{3} (x_{i}), & array x \in [x_{i}, x_{i + 1}], \\ array y (x_{i}) (y (x_{i}) z (x_{i + 2}) + z (x_{i + 3}) y (x_{i + 1})) + z (x_{i + 4}) y^{2} (x_{i + 1}), & array x \in [x_{i + 1}, x_{i + 2}], \\ array ... \end{matrix})

…”

Section: Derivation Of the Schemementioning

confidence: 99%

Numerical simulation of the nonlinear generalized time‐fractional Klein–Gordon equation using cubic trigonometric B‐spline functions

Yaseen

Abbas

Ahmad

2020

Math Methods in App Sciences

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In this paper, an efficient numerical procedure for the generalized nonlinear time-fractional Klein-Gordon equation is presented. We make use of the typical finite difference schemes to approximate the Caputo time-fractional derivative, while the spatial derivatives are discretized by means of the cubic trigonometric B-splines. Stability and convergence analysis for the numerical scheme are discussed. We apply our scheme to some typical examples and compare the obtained results with the ones found by other numerical methods. The comparison shows that our scheme is quite accurate and can be applied successfully to a variety of problems of applied nature.

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“…There are a large number of studies related to B-splines for the solutions of FPDEs but only limited studies dealt to the solution of nonlinear KG equations. [25][26][27][28][29] Yaseen et al 30 solved time fractional diffusion wave equation numerically using trigonometric B-splines technique. Fractional Predator-Prey models have been numerically solved by Pitolli 31 using fractional Bspline approach.…”

Section: Introductionmentioning

confidence: 99%

Development and Analysis of New Approximation of Extended Cubic B-Spline to the Nonlinear Time Fractional Klein–gordon Equation

et al. 2020

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A new extended cubic B-spline (ECBS) approximation is formulated, analyzed and applied to obtain the numerical solution of the time fractional Klein–Gordon equation. The temporal fractional derivative is estimated using Caputo’s discretization and the space derivative is discretized by ECBS basis functions. A combination of Caputo’s fractional derivative and the new approximation of ECBS together with [Formula: see text]-weighted scheme is utilized to obtain the solution. The method is shown to be unconditionally stable and convergent. Numerical examples indicate that the obtained results compare well with other numerical results available in the literature.

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An efficient computational technique based on cubic trigonometric B-splines for time fractional Burgers' equation

Cited by 35 publications

References 38 publications

A numerical algorithm based on modified extended B-spline functions for solving time-fractional diffusion wave equation involving reaction and damping terms

A numerical algorithm based on modified extended B-spline functions for solving time-fractional diffusion wave equation involving reaction and damping terms

Numerical simulation of the nonlinear generalized time‐fractional Klein–Gordon equation using cubic trigonometric B‐spline functions

Development and Analysis of New Approximation of Extended Cubic B-Spline to the Nonlinear Time Fractional Klein–gordon Equation

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