Some Higher-Degree Lacunary Fractional Splines in the Approximation of Fractional Differential Equations

Srivástava, H. M.; Mohammed, Pshtiwan Othman; Guirao, Juan Luis García; Hamed, Y. S.

doi:10.3390/sym13030422

Cited by 12 publications

(6 citation statements)

References 17 publications

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“…Kadalbajoo and Arora [54] established a B-spline collocation methodology that solves singular-perturbed equation using artificial viscosity. Convergence of odd degree equation and error bounds for spline interpolation presented in [55][56][57].…”

Section: Introductionmentioning

confidence: 99%

Extension of Cubic B-Spline for Solving the Time-Fractional Allen–Cahn Equation in the Context of Mathematical Physics

Fatima,

Agarwal,

Abbas

et al. 2024

Computation

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A B-spline is defined by the degree and quantity of knots, and it is observed to provide a higher level of flexibility in curve and surface layout. The extended cubic B-spline (ExCBS) functions with new approximation for second derivative and finite difference technique are incorporated in this study to solve the time-fractional Allen–Cahn equation (TFACE). Initially, Caputo’s formula is used to discretize the time-fractional derivative, while a new ExCBS is used for the spatial derivative’s discretization. Convergence analysis is carried out and the stability of the proposed method is also analyzed. The scheme’s applicability and feasibility are demonstrated through numerical analysis.

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Section: Introductionmentioning

confidence: 99%

Extension of Cubic B-Spline for Solving the Time-Fractional Allen–Cahn Equation in the Context of Mathematical Physics

Fatima,

Agarwal,

Abbas

et al. 2024

Computation

Self Cite

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“…The main advantage of fractional order differential equations is that they produce accurate and stable results. Therefore, these equations represent a significant class of differential equations [11][12][13][14][15][16][17][18].…”

Section: Introductionmentioning

confidence: 99%

An efficient approach to solving fractional Van der Pol–Duffing jerk oscillator

El‐Dib¹

2022

Commun. Theor. Phys.

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The motive behind the current work is to perform the solution of the Van der Pol–Duffing jerk oscillator, involving fractional-order by the simplest method. An effective procedure has been introduced for executing the fractional-order by utilizing a new method without the perturbative approach. The approach depends on converting the fractional nonlinear oscillator to a linear oscillator with an integer order. The sincerity of the established results is justified by providing a suitable example of nonlinear oscillators. The obtained analytical solutions are supported by the numerical simulation results. The method applied here does not require hard mathematical work, which makes it distinct from other methods. The simplicity of the present approach provides extra advantages for fractional-order solutions. This method enriches the analysis with more details in new dimensions.

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“…Fractional differentiation and integration have opened many new doors for researchers in recent decades due to their wide and novel applicability in many fields of science including mathematical analysis, technology, and engineering (see [1][2][3][4][5][6][7]). Many techniques are used to deal with these new differential and integral operators; for instance, some researchers used analytical techniques including Laplace transform, spline interpolation, Green function, Crank-Nicolson approximation method, method of separation of variable, and many others to derive exact solutions to linear differential or integral equations (see [8][9][10][11][12][13][14]). Using the fixed-point technique, some researchers provided the conditions under which differential and integral equations have unique solutions.…”

Section: Introductionmentioning

confidence: 99%

On Riemann—Liouville and Caputo Fractional Forward Difference Monotonicity Analysis

2021

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Monotonicity analysis of delta fractional sums and differences of order υ∈(0,1] on the time scale hZ are presented in this study. For this analysis, two models of discrete fractional calculus, Riemann–Liouville and Caputo, are considered. There is a relationship between the delta Riemann–Liouville fractional h-difference and delta Caputo fractional h-differences, which we find in this study. Therefore, after we solve one, we can apply the same method to the other one due to their correlation. We show that y(z) is υ-increasing on Ma+υh,h, where the delta Riemann–Liouville fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to zero, and then, we can show that y(z) is υ-increasing on Ma+υh,h, where the delta Caputo fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to −1Γ(1−υ)(z−(a+υh))h(−υ)y(a+υh) for each z∈Ma+h,h. Conversely, if y(a+υh) is greater or equal to zero and y(z) is increasing on Ma+υh,h, we show that the delta Riemann–Liouville fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to zero, and consequently, we can show that the delta Caputo fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to −1Γ(1−υ)(z−(a+υh))h(−υ)y(a+υh) on Ma,h. Furthermore, we consider some related results for strictly increasing, decreasing, and strictly decreasing cases. Finally, the fractional forward difference initial value problems and their solutions are investigated to test the mean value theorem on the time scale hZ utilizing the monotonicity results.

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Some Higher-Degree Lacunary Fractional Splines in the Approximation of Fractional Differential Equations

Cited by 12 publications

References 17 publications

Extension of Cubic B-Spline for Solving the Time-Fractional Allen–Cahn Equation in the Context of Mathematical Physics

Extension of Cubic B-Spline for Solving the Time-Fractional Allen–Cahn Equation in the Context of Mathematical Physics

An efficient approach to solving fractional Van der Pol–Duffing jerk oscillator

On Riemann—Liouville and Caputo Fractional Forward Difference Monotonicity Analysis

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