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Numerical Solutions with Linearization Techniques of the Fractional Harry Dym Equation

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Asif Yokus at Fırat University
Asif Yokus
Sema Gülbahar
Sema Gülbahar
Sema Gülbahar
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Abstract

In this study, numerical solutions of the fractional Harry Dym equation are investigated. Linearization techniques are utilized for non-linear terms existing in the fractional Harry Dym equation. The error norms L2 and L∞ are computed. Stability of the finite difference method is studied with the aid of Von Neumann stabity analysis.
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Open Access. © 2019 Asıf Yoku¸s and Sema Gülbahar, published by Sciendo.
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Applied Mathematics and Nonlinear Sciences 4(1) (2019) 35–42
Applied Mathematics and Nonlinear Sciences
https://www.sciendo.com
Numerical Solutions with Linearization Techniques of the Fractional Harry Dym
Equation
Asıf Yoku¸s1, Sema Gülbahar2
1Department of Actuary, Firat University, 2300, Elazig, Turkey,
E-mail: asfyokus@firat.edu.tr
2Department of Mathematics, Harran University, Sanliurfa, Turkey,
E-mail: semaakkus_mat@hotmail.com
Submission Info
Communicated by Juan Luis García Guirao
Received 14th January 2019
Accepted 21st February 2019
Available online 27th March 2019
Abstract
In this study, numerical solutions of the fractional Harry Dym equation are investigated. Linearization techniques are
utilized for non-linear terms existing in the fractional Harry Dym equation. The error norms L2and L∞are computed.
Stability of the finite difference method is studied with the aid of Von Neumann stabity analysis.
Keywords: Harry Dym equation, finite difference method, von Neumann stability analysis.
1 Introduction
Fractional analysis; is the generalization of the classic analysis of integration and differentiation of process
(noninteger) order. This issue is an old issue as much as differential calculus. G. W. Leibniz and Marquis de
L0Hospital correspondence in 1695, is known as the first exit point of fractional calculus. Leibniz expressed
fractional order derivatives of noninteger order f(x) = emx,m∈R, as follows:
dnemx
dxn=mnemx ,
here, nis a value, where i is the noninteger.
Later, many scientists, such as Liouville, Riemann, Weyl, Lacroix, Leibniz, Grunward and Letnikov (cf. [4]),
expanded the range of this derivative.
Since the beginning of the definition of fractional order derivatives first handled by Leibniz, fractional partial
differential equations have attracted the attention of many scientists and have also shown a progressive develop-
ment.(cf. [1–9,11,13–19]).
ISSN 2444-8656 doi:10.2478/AMNS.2019.1.00004
36 J.P. Ruiz-Fernández et al. Applied Mathematics and Nonlinear Sciences 4(2019) 35–42
Beside these facts, the third order fractional Harry Dym partial differential equation is studied in mathematics
and especially in the theory of solitons. This equation is given as
∂γu
∂tγ=u3∂3u
∂x3.(1.1)
Here, 0 <γ≤1 is the order of fractional derivative and u(x,t)is a function of xand t.
The Harry Dym equation first appeared in a study by Kruskal [10]. Harry Dym equation represents a system
in which dispersion and nonlinearity were coupled. Furthermore, the Harry Dym equation is a completely
integrable nonlinear evolution that may be solved by means of the inverse scattering transform. It does not
possess the Painlevé property.
This paper is organized as follows:
In the second section, some basic facts dealing with the finite difference method are mentioned and three Lin-
earization techniques are presented. In the third section, stability analysis of the proposed method is investigated
and it is shown that the Harry Dym equation is stable under which conditions. Also, numerical examples are
given. In the fourth section, conclusions obtained throughout the paper are discussed.
2 Finite Difference Methods
In this section, we first need to define a set of grid points in a domain Dto obtain a numerical solution to Eq.
(1.1)using finite difference methods as follows:
Let ∆x(h) = b−a
N(Nis an integer) denotes a state step size and ∆tdenotes a time step size. Draw a set of
horizontal and vertical line across D, and get all intersection points (xj,tn)or simply (j,n)where xj=a+j∆x,
j=0,1,2,··· ,N, and tn=n∆t,n=0,1,...,M. If we write D= [a,b]×[0,T]then we may choose ∆t=T
M(M
is an integer) and tn=n∆t,n=0,1,...,M.
Then, an appropriate finite difference approximation is given in Eq. (1.1) instead of derivatives and its vari-
able. In this case, the solution problem of Eq. (1.1) is reduced to solution problem of algebraic differential sys-
tems of linear and nonlinear equations consisting of finite difference equation. But when applied to non-linear
problems, it normally leads to nonlinear system of equations and they cannot be solved directly. Therefore, we
use three linearization techniques for a nonlinear term as given in Eq. (1.1).
2.1 Linearization 1
First, we use the Caputo fractional derivative approximation for ∂γu
∂tγdefined by
∂γu
∂tγ'1
h2
(∆t)−γ
Γ(2−γ)
n
∑
k=0un+1−k
m−un−k
mh(k+1)1−γ−k1−γin≥1
(∆t)−γ
Γ(2−γ)u1
m−u0
mn=0,
and Crank–Nicolson derivative approximation given by
u=un+1+un
2,
in Eq. (1.1) at the nodal point (m,n+1)[12]. Then, if we discretize time derivative of the fractional Harry Dym
equation by using Caputo fractional derivative formula between two successive time levels nand n+1, Crank
Nicolson derivative formula and usual finite difference formula between two successive time levels nand n+1,
Numerical Solutions with Linearization Techniques of the Fractional Harry Dym Equation 37
respectively, we obtain
(∆t)−γ
Γ(2−γ)
n
∑
k=0hun+1−k
m−un−k
mih(k+1)1−γ−k1−γi(2.1)
="u3n+1
m+u3n
m
2#un
m+2−2un
m+1+2un
m−1−un
m−2
2h3.
The nonlinear term in the above mentioned equation is linearized by using the following equation:
u3n+1
m=3u2nun+1−2u3n.
If the necessary arrangements are made in Eq. (2.1), we have the following equation:
(∆t)−γ
Γ(2−γ)
n
∑
k=0hun+1−k
m−un−k
mih(k+1)1−γ−k1−γi(2.2)
=1
4h3h3u2n
mun+1
m−u3niun
m+2−2un
m+1+2un
m−1−un
m−2.
2.1.1 Linearization 2
Let us use the Caputo fractional derivative approximation for ∂γu
∂tγ., Crank–Nicolson derivative approximation
and usual finite difference approximation for Uxxx given by
Uxxx '1
2h3un
m+2−2un
m+1+2un
m−1−un
m−2,
in Eq. (1.1) at the nodal point (m,n+1), respectively [12]:
If we use the following linearization technique for the non-linear term U3Uxxx:
U3Uxxx '(Un
m)31
2h3un
m+2−2un
m+1+2un
m−1−un
m−2,
then we have the following system of algebraic equation:
(∆t)−γ
Γ(2−γ)
n
∑
k=0hun+1−k
m−un−k
mih(k+1)1−γ−k1−γi(2.3)
= (Un
m)31
2h3un+1
m+2−2un+1
m+1+2un+1
m−1−un+1
m−2.
Eq. (2.3) can be solved using an approximate algorithm.
2.1.2 Linearization 3
Let us use the Caputo fractional derivative approximation for ∂γu
∂tγand usual finite difference approximation
for Uxxx in Eq. (1.1) at the nodal point (m,n+1)respectively [12]: If we use the following linearization technique
for the nonlinear term U3Uxxx, then we have
U3Uxxx '(Un
m+Un
m+1
2)31
2h3un
m+2−2un
m+1+2un
m−1−un
m−2.
Thus, we get the following system of algebraic equation:.
(∆t)−γ
Γ(2−γ)
n−1
∑
k=0hun−k
m−un−k−1
mih(k+1)1−γ−k1−γi(2.4)
= (Un
m+Un
m+1
2)31
2h3un
m+2−2un
m+1+2un
m−1−un
m−2.
Eq. (2.4)can be solved using an approximate algorithm.
38 J.P. Ruiz-Fernández et al. Applied Mathematics and Nonlinear Sciences 4(2019) 35–42
3 Stability analysis
In this section, we investigate whether this method is stable based on von-Neumann analysis. If the Fourier
method analyzes the stability, then the growth factor of a typical Fourier mode is defined as:
un
m=ξne`mϕ, ` =√−1,(3.1)
where ξnis considered as the amplification factor. First, by substituting the Fourier mode (3.1)into the recur-
rence relationship (2.3), one can obtain
4h3(∆t)−γ
Γ(2−γ)
n−1
∑
k=0h(k+1)1−γ−k1−γihξn−k+1−ξn−kie`mϕ
= (ξne`mϕ)3ξn+1e`(m+2)ϕ−2ξn+1e`(m+1)ϕ+2ξn+1e`(m−1)ϕ−ξn+1e`(m−2)ϕ.(3.2)
Next, if we assume that ξn+1=ζ ξ nand ζ=ζ(ϕ)are independent of time, we can easily obtain the
following expression:
4h3(∆t)−γ
Γ(2−γ)
n−1
∑
k=0h(k+1)1−γ−k1−γihζ−k−3n+1−ξ−k−3ni
=ξe3`mϕe2`ϕ−2e`ϕ+2e−`ϕ−e−2`ϕ.(3.3)
Hence, we get
ξ=X1+iX2
Y1+iY2
,(3.4)
X1=2(∆t)−γ
Γ(2−γ)
n−1
∑
k=0h(k+1)1−γ−k1−γihζ1−3n−k−ζ−3n−ki,
X2=0,
Y1=−24cos2[mγ]sin [mϕ]sin2hϕ
2isin[ϕ] + 8 sin3[mϕ]sin2hϕ
2isin[ϕ],
Y2=−24cos [mγ]sin2[mϕ]sin2hϕ
2isin[ϕ] + 8 cos3[mϕ]sin2hϕ
2isin[ϕ],(3.5)
which shows that
|ζ|=
X1+X2
Y1+Y2
.(3.6)
For the Fourier stability definition and for the examined scheme to be stable, the condition |ξ|≤1 must be
gratified. Therefore, if the following inequality is provided, the schema is unconditionally stable.
X2
1
Y2
1+Y2
2≤1.(3.7)
Other schemes can be studied by following a similar way.
3.1 L2and L∞error norms
The equation of the numerical results was obtained for the test problem used in this study and all computa-
tions have been run on using double precision arithmetic. To show how accurate the results, both the error norm
L2
Numerical Solutions with Linearization Techniques of the Fractional Harry Dym Equation 39
L2=
Uexact −UN
2=sh
N
∑
J=0Uexact
j−(UN)j
2
,
and L∞
L∞=
Uexact −UN
∞=max
jUexact
j−(UN)j,
are going to be computed and presented.
3.2 Test Problem
The analytical solution of the fractional Harry Dym equation is given as follows [11].
u(x,t) = 4−3
2(x+t)2
3
,t≥t0,0≤x≤1.(3.8)
In our computations, for the numerical solution of the test problem three different linearization techniques
have been applied. The values of the error norms L2and L∞have been computed at t=1 for different values of
∆t,h. The comparison of the error norms L2and L∞obtained by the linearization techniques is summarized in
Table 1. As summarized in Table 1, it is obvious that the obtained results using linearization 1 are better than the
obtained results using other linearizations. As shown in Figure 1, one can compare the exact and approximate
solution of the collocation method using four radial basis functions for h=0.01.Similarly, as shown in Figures
2 and 3 the exact and approximate solutions can be compared for linearization 2 and linearization 3 successively.
Table 1 Comparison of the error norms L2and L∞that are obtained using the linearization techniques at t=1,α=0.9
for different values of ∆t,h.
Lin. I Lin. II Lin. III
L2×102L∞×102L2×102L∞×102L2×102L∞×102
4t=h=0.001 0.00002 0.00002 0.00566 0.05664 0.00565 0.05659
4t=h=0.05 0.05254 0.08344 0.64556 0.94172 0.58516 0.84499
4t=h=0.04 0.02957 0.05128 0.50690 0.82106 0.47224 0.76100
4t=h=0.03 0.01418 0.02773 0.36800 0.68391 0.35113 0.65086
4t=h=0.02 0.00506 0.01186 0.23124 0.52313 0.22502 0.50855
4t=h=0.01 0.00088 0.00285 0.10161 0.32318 0.10046 0.31945
4t=h=0.2 2.44568 0.00285 3.12756 3.78842 0.80756 1.29370
4t=h=0.1 3.27620 0.41645 1.31211 1.41287 0.94027 0.94563
4 Conclusions
Finite difference methods based on using three different linearization techniques have been proposed for the
numerical solutions of the fractional Harry Dym equation. In addition, numerical results were obtained by using
three different linearization techniques. The proposed method has been tested on a problem and demonstrated
how effective it is. The error norms L2and L∞have been calculated and given. The third linearization technique,
as shown in Figure 3, yielded better results. Because the third linearization technique gives better results, this
technique can be suggested in the next problems and the situation of the problem should be considered. The
obtained results show that the error norms are sufficiently small during all computer runs. It has been observed
that the considered method is a power numerical scheme to solve the fractional Harry Dym equation.
40 J.P. Ruiz-Fernández et al. Applied Mathematics and Nonlinear Sciences 4(2019) 35–42
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
x
0
0.5
1.0
1.5
2.0
2.5
u(x,t)
0.460 0.473 0.486 0.500
x
1.40
1.45
1.50
1.55
u(x,t)
Exact Solution
Approximate Solution
Fig. 1 Comparison of the exact and approximate solution for linearization 2 at h=0.01
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
x
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
u(x,t)
0.465 0.475 0.485 0.495
x
2.195
2.200
2.205
2.210
2.215
u(x,t)
Exact Solution
Approximate Solution
Fig. 2 Comparison of the exact and approximate solution for linearization 3 at h=0.01
References
[1] Al-Khaled, K. and Alquran, M. (2014). An approximate solution for a fractional model of generalized Harry Dym
equation. Mathematical Sciences, 8(4), 125-130.
[2] Biswas, A., Mirzazadeh, M., Eslami, M., Milovic, D., & Belic, M. (2014). Solitons in optical metamaterials by func-
tional variable method and first integral approach. Frequenz, 68(11-12), 525-530.
[3] Baskonus, H., Mekkaoui, T., Hammouch, Z., and Bulut, H. (2015). Active control of a chaotic fractional order eco-
nomic system. Entropy, 17(8), 5771-5783.
[4] Das, S. (2011). Functional fractional calculus. Springer Science & Business Media.
[5] Esen, A., Ucar, Y., Yagmurlu, N. and Tasbozan, O. (2013). A Galerkin finite element method to solve fractional
Numerical Solutions with Linearization Techniques of the Fractional Harry Dym Equation 41
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
x
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
u(x,t)
0.596 0.603 0.610 0.616
x
2.02
2.03
2.04
2.05
2.06
2.07
u(x,t)
Exact Solution
Approximate Solution #1
Approximate Solution #2
Approximate Solution #3
Fig. 3 Comparison of the exact and approximate solution for linearization 1, linearization 2, and linearization 3 at
h=0.01
diffusion and fractional diffusion-wave equations. Mathematical Modelling and Analysis, 18(2), 260-273.
[6] Ravichandran, C., Jothimani, K., Baskonus, H. M., and Valliammal, N. (2018). New results on nondensely character-
ized integrodifferential equations with fractional order. The European Physical Journal Plus, 133(3), 109.
[7] Kaya, D., Gülbahar, S., Yoku¸s, A. and Gülbahar, M. (2018). Solutions of the fractional combined KdV–mKdV equa-
tion with collocation method using radial basis function and their geometrical obstructions. Advances in Difference
Equations, 2018(1), 77.
[8] Kumar, D., Singh, J. and Kılıçman, A. (2013). An efficient approach for fractional Harry Dym Equation by Using
Sumudu Transform, Abstr. Appl. Anal., 2013, Article ID 608943, 8.
[9] Kumar, S., Tripathi, M.P. and Singh, O.P. (2013). A fractional model of Harry Dym equation and its approximate
solution, Ain Shams Eng. J., 4 , 111-115.
[10] Kruskal, M. (1975). Nonlinear Wave Equations. In Jürgen Moser, editor, Dynamical Systems, Theory and Applica-
tions,Heidelberg. Springer. 38, 310–354.
[11] Rawashdeh, M., S. (2014). New approach to solve the fractional Harry Dym equation using the FRDTM, Int. J. Pure
Appl. Math., 95, 553-566.
[12] Kaya, D., Yoku¸s, A. ( 2014). Stability Analysis and Numerical Solutions for Time Fractional KdVB Equation, Interna-
tional Comference on Computational Experimental Science and Engineering, Antalya.
[13] Esen, A., Sulaiman, T. A., Bulut, H., and Baskonus, H. M. (2018). Optical solitons to the space-time fractional (1+
1)-dimensional coupled nonlinear Schr dinger equation. Optik, 167, 150-156.
[14] Yang, X. J., Gao, F., Machado, J. A., and Baleanu, D. (2017). A new fractional derivative involving the normalized sinc
function without singular kernel. arXiv preprint arXiv:1701.05590.
[15] Yavuz, M., Ozdemir, N. and Baskonus, H. M. (2018). Solutions of partial differential equations using the fractional
operator involving Mittag-Leffler kernel. The European Physical Journal Plus, 133(6), 215.
[16] Yavuz, M. (2019). Characterizations of two different fractional operators without singular kernel. Mathematical Mod-
elling of Natural Phenomena, 14(3), 302.
[17] Bulut, H., Kumar, D., Singh, J., Swroop, R., and Baskonus, H. M. (2018). Analytic study for a fractional model of HIV
infection of CD4+ TCD4+ T lymphocyte cells. Math. Nat. Sci, 2(1), 33-43.
[18] Baskonus, H. M., Hammouch, Z., Mekkaoui, T., and Bulut, H. (2016, June). Chaos in the fractional order logistic delay
system: Circuit realization and synchronization. In AIP Conference Proceedings (Vol. 1738, No. 1, p. 290005). AIP
Publishing.
[19] Bulut, H., Yel, G., Ba konu , H. M. (2016). An application of improved Bernoulli sub-equation function method to
the nonlinear time-fractional burgers equation. Turkish Journal of Mathematics and Computer Science, 5, 1-7.
42 J.P. Ruiz-Fernández et al. Applied Mathematics and Nonlinear Sciences 4(2019) 35–42
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This study is centered on addressing the complexities of the nonlinear time-fractional Harry Dym equation through a combined application of analytical and numerical methodologies. The principal objective is to establish a comprehensive framework for solving this intricate fractional partial differential equation. The introduction of the Khater II method as an analytical technique presents a novel approach specifically designed to handle such equations. In tandem, the utilization of numerical schemes-cubic-B-spline, quantic-B-spline, and septic-B-spline methods-further refines the accuracy and efficiency of the solutions. The outcomes underscore the efficacy of these proposed methods in effectively resolving the challenges posed by the Harry Dym equation. Emphasis is placed on the dual facets of analytical insights and numerical precision. The significance of this study lies in its contribution to advancing comprehension of nonlinear time-fractional equations while offering practical tools for their resolution. The conclusions drawn from this research provide valuable insights into the practical applicability of the Khater II method and B-spline schemes in navigating complex mathematical models. The novelty of this work lies in the amalgamation of an innovative analytical approach with established numerical techniques, signifying a noteworthy contribution to this field of study.
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... Numerical, analytical or accurate solutions of partial differential equations have been one of the most exciting fields of research for many scholars around the world for several decades. Many methods are utilized to construct and study numerical or analytical solutions such as, the Adomian decomposition method (Ismael and Ali 2017;Bulut et al. 2004), the Adams-Bashforth-Moulton method (Veeresha et al. 2022), the improved shooting method (Javeed and Hincal 2024), the finite forward difference method (Yokuş and Gülbahar 2019), the homotopy analysis method (Veeresha et al. 2021a, b), the Haar wavelet method (Oruç et al. 2016), the homotopy perturbation method (Yousif et al. 2016), the generalized Langevin equation method (Ness et al. 2015), the Laplace transformation method (Häser and Almlöf 1992), improved sub-Bernoulli equation (Baskonus and Bulut 2016), the modified exp (− ( ))-expansion function method (Cattani et al. 2018;Houwe et al. 2019;Ilhan et al. 2018;Akram and Sajid 2021), Hirota method (Ismael et al. 2021(Ismael et al. , 2022, the sine-Gordon expansion method (Ciancio et al. 2022;Kumar et al. 2021), the tan ( ( )∕2)-expansion technigue (Hammouch et al. 2018;Manafian et al. 2016), the improved tan ( ( )∕2)-expansion technigue , extended rational sinh-Gordon method (Rehman and Ahmad 2023a, b), reductive perturbation technique (Seadawy et al. 2020b;Seadawy and Iqbal 2021), extended modified rational expansion method (Iqbal et al. 2020(Iqbal et al. , 2022(Iqbal et al. , 2023a(Iqbal et al. , b, 2024, the Darboux transformation method (Guo et al. 2014;He et al. 2012), the modified form of Kudryashov approach (Hosseini et al. 2018;Kabir et al. 2011), the Riccati-Bernoulli sub-ODE method (Yang et al. 2015; Abdelrahman and Sohaly 2021), G � ∕G -expansion method (Aghdaei and Manafian 2016;Khalique and Mhlanga 2018), the modified auxiliary expansion method Akram et al. 2023aAkram et al. , 2022bFaridi et al. 2023b), extended direct algebraic technique (Tarla et al. 2022;Faridi et al. 2024;Asjad et al. 2023;Hussain et al. 2023), the generalized projective Riccati equation approach (Akram et al. 2022a;Majid et al. 2023), 1 G ′ -expansion method (Yokuş 2018;Ali et al. 2022a), the generalized exponential rational function method (GERFM) (Osman and Ghanbari 2018;Ali et al. 2021b;Rehman et al. 2022;Ahmad 2023), a modified GERFM Ahmad et al. 2023), the Φ 6 -model expansion method , the extended G � ∕G 2 -expansion technigue (Akram et al. 2023b(Akram et al. , 2022cFaridi et al. 2023a), modified extended auxiliary equation mapping method (Lu et al. 2018a, b), Lie symmetry approach (Faridi and AlQahtani 2023;Faridi 2023), the modified Sardar-sub equation ) and many other techniques (Seadawy and Iqbal 2023;Seadawy et al. , 2020aChen et al. 2022). ...
Novel optical solutions to the dispersive extended Schrödinger equation arise in nonlinear optics via two analytical methods
Article
Full-text available
  • Mar 2024
  • OPT QUANT ELECTRON
The main goal of this paper is to study the higher-order dispersive extended nonlinear Schrödinger equation, which demonstrates the propagation of ultrashort pulses in optical communication networks. In this study, both the sinh-Gordon expansion method and the generalized exponential rational function method are used to offer some novel optical solutions. These optical soliton solutions are dark soliton, bright soliton, singular, periodic, and dark-bright soliton solutions. The obtained optical soliton solutions are presented graphically in 2D and 3D to clarify the behavior of solutions more effectively. The constraint conditions are also used to verify the exitances of the new analytical solutions. Moreover, all solutions compared to solutions obtained previously are new, and all the new wave solutions have verified Eq. (1) after we substituted them into the studied equation. In the future, these novel soliton solutions will be very helpful in developing fluid dynamics, biomedical issues, dynamics of adiabatic parameters, industrial research, and many other areas of science. To our acknowledgment, the presented optical solutions are novel, and also beforehand these methods have not been applied to this studied equation.
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... They developed a number of strategies, methodologies, and theories for 118 computing precise or numerical FODE solutions. For the desired results, researchers 119 employ well-known techniques such as HPM, ADM and a variety of other numerical 120 methods see [47][48][49]. Motivated by this work, in our project we handle the model 121 (1) under CFFD as 123 where H 0 , V 0 , U 0 , A 0 , C 0 ≥ 0 and σ ∈ (0, 1]. Firstly some qualitative results, such 124 as the existence and uniqueness of the solution that corresponds to the given model, 125 are proven. ...
Qualitative Theory and Approximate Solution to Norovirus Model Under Non Singular Kernel Type Derivatives
Chapter
Full-text available
  • Oct 2023
We study a fractional order nonlinear dynamical system of Norovirus disease. The considered mathematical model consists of susceptible, exposed, vaccinated, infected, and recovered classes. Firstly, the Caputo-Fabrizio fractional derivative (CFFD) is used to examine the qualitative theory and approximate solutions. Some adequate results for the existence of approximate solutions are given using fixed point theory. We use the Laplace Transform (LT) and the Adomian decomposition (ADM) approach to get approximate results for each compartment. The graphical presentation that corresponds to available real facts for different fractional orders is given. Secondly, by using piecewise global fractional derivatives in sense of singular and non-singular kernels, the aforementioned system is also investigated. We establish sufficient conditions for the existence and uniqueness of the solution to the proposed model by using the fixed-point approach. Numerical simulation is performed by extending Newton’s interpolation formula for the considered model under the mentioned derivatives. Graphical presentations for various compartments of the model are given against the available real data for different fractional orders. The numerical findings are performed by using Matlab 16.
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Stochastic Performances of the Mathematical System Based on the Prevalence Prediction of Allergies
Article
  • Jun 2024
The objective of this study is to conduct numerical simulations of a six-compartment model in order to predict the likelihood of developing allergies. The research categorizes the division of model into four classes of human population, along with two allergen groups, namely inhalants and food. The study aims to develop a predictive model for allergen sensitization occurrence using a system of nonlinear differential equations. The numerical presentation of the system's solution is achieved through the utilization of the stochastic computational artificial neural network (ANN) and the Levenberg–Marquardt backpropagation (LMBP). The neural network process involves presenting three types of statistics: training, testing, and sampling. The allergy occurrence prediction model has a preferred statistical accuracy of 72% for training data and 14% for both authentication and testing data. The accuracy of the stochastic scheme is evaluated by comparing its performance with the proposed and reference solutions. A small absolute error is an indication of the scheme's correctness. The research observes the reliability of the stochastic solver by utilizing various statistical operators.
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NONLINEARITY AND MEMORY EFFECTS: THE INTERPLAY BETWEEN THESE TWO CRUCIAL FACTORS IN THE HARRY DYM MODEL
Article
  • May 2024
This study investigates the nonlinear time-fractional Harry Dym ([Formula: see text]) equation, a model with significant applications in soliton theory and connections to various other nonlinear evolution equations. The Harry Dym ([Formula: see text]) equation describes the propagation of nonlinear waves in various physical contexts, including shallow water waves, nonlinear optics, and plasma physics. The fractional-order derivative introduces a memory effect, allowing the model to capture nonlocal interactions and long-range dependencies in the wave dynamics. The primary objective of this research is to obtain accurate analytical solutions to the [Formula: see text] equation and explore its physical characteristics. We employ the Khater III method as the primary analytical technique and utilize the He’s variational iteration ([Formula: see text]) method as a numerical scheme to validate the obtained solutions. The close agreement between analytical and numerical results enhances the applicability of the solutions in practical applications of the model. This research contributes to a deeper understanding of the [Formula: see text] equation’s behavior, particularly in the presence of fractional-order dynamics. The obtained solutions provide valuable insights into the complex interplay between nonlinearity and memory effects in the wave propagation phenomena described by the model. By shedding light on the physical characteristics of the [Formula: see text] equation, this study paves the way for further investigations into its potential applications in diverse physical settings.
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Existence and Stability Behaviour of FSDE Driven by Rosenblatt Process with the Application of Visual Perception of Fish Robot
Article
Full-text available
  • Feb 2024
The successive approximation used to derive the existence and stability results of the fractional stochastic differential equation (FSDEs) driven by the Rosenblatt process and numerical simulation are established and applied for the reduction of stochastic disturbance of minimal level in the visual perception trajectory. The Rosenblatt process ensures the stability of FDSEs by mitigating the stochastic disruption in the ocean water environment, including small particles along the visual perception trajectory to the fish robot. The algorithms have several advantages from gaze shift frames, such as terrific quality of randomness, key sensitivity, and minimizing the stochastic disturbance in the visual perception track for different locations. Numerical simulation results manifest real-world applications’ effectiveness, efficiency, and feasibility.
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Solutions of the fractional combined KdV–mKdV equation with collocation method using radial basis function and their geometrical obstructions
Article
Full-text available
  • Mar 2018
  • ADV DIFFER EQU-NY
The exact solution of fractional combined Korteweg-de Vries and modified Korteweg-de Vries (KdV–mKdV) equation is obtained by using the (1/G′)(1/G^{\prime}) expansion method. To investigate a geometrical surface of the exact solution, we choose γ=1\gamma=1. The collocation method is applied to the fractional combined KdV–mKdV equation with the help of radial basis for 0<γ<10<\gamma<1. L2L_{2} and L∞L_{\infty} error norms are computed with the Mathematica program. Stability is investigated by the Von-Neumann analysis. Instable numerical solutions are obtained as the number of node points increases. It is shown that the reason for this situation is that the exact solution contains some degenerate points in the Lorentz–Minkowski space.
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Active Control of a Chaotic Fractional Order Economic System
Article
Full-text available
In this paper, a fractional order economic system is studied. An active control technique is applied to control chaos in this system. The stabilization of equilibria is obtained by both theoretical analysis and the simulation result. The numerical simulations, via the improved Adams–Bashforth algorithm, show the effectiveness of the proposed controller.
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New results on nondensely characterized integrodifferential equations with fractional order
Article
  • Mar 2018
This study reports a novel approach which deals with the coupled classes of fractional integrodifferential equations for nondensely characterized linear operators in the Banach space. Using the noncompact measure theory, we investigate the existence of the results presented.
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Characterizations of Two Different Fractional Operators without Singular Kernel
Article
  • Feb 2019
In this paper, we analyze the behaviours of two different fractional derivative operators defined in the last decade. One of them is defined with the normalized sinc function (NSF) and the other one is defined with the Mittag-Leffler function (MLF). Both of them have a non-singular kernel. The fractional derivative operator defined with the MLF is developed by Atangana and Baleanu (ABO) in 2016 and the other operator defined with the normalized sinc function (NSFDO) is created by Yang et al. in 2017. These mentioned operators have some advantages to model the real life problems and to solve them. On the other hand, since the Laplace transform (LT) of the ABO can be calculated more easily, it can be preferred to solve linear/nonlinear problems. In this study, we use the perturbation method with coupled the LTs of these operators to analyze their performance in solving some fractional differential equations. Furthermore, by constructing the error analysis, we test the practicability and usefulness of the method.
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Solutions of partial differential equations using the fractional operator involving Mittag-Leffler kernel
Article
  • Jun 2018
In this paper, time-fractional partial differential equations (FPDEs) involving singular and non-singular kernel are considered. We have obtained the approximate analytical solution for linear and non-linear FPDEs using the Laplace perturbation method (LPM) defined with the Liouville-Caputo (LC) and Atangana-Baleanu (AB) fractional operators. The AB fractional derivative is defined with the Mittag-Leffler function and has all the properties of a classical fractional derivative. In addition, the AB operator is crucial when utilizing the Laplace transform (LT) to get solutions of some illustrative problems with initial condition. We show that the mentioned method is a rather effective and powerful technique for solving FPDEs. Besides, we show the solution graphs for different values of fractional order α,d i s t a n c e term x and time value t. The classical integer-order features are fully recovered if α is equal to 1.
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A new fractional derivative involving the normalized sinc function without singular kernel
Article
  • May 2018
In this paper, a new fractional derivative involving the normalized sinc function without singular kernel is proposed. The Laplace transform is used to find the analytical solution of the anomalous heat-diffusion problems. The comparative results between classical and fractional-order operators are presented. The results are significant in the analysis of one-dimensional anomalous heat-transfer problems.
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Chaos in the fractional order logistic delay system: Circuit realization and synchronization
Conference Paper
  • Jun 2016
Articles you may be interested in Bifurcation behaviors of synchronized regions in logistic map networks with coupling delay Chaos 25, 033101 (2015); 10.1063/1.4913854 Synchronization and an application of a novel fractional order King Cobra chaotic system Chaos 24, 033105 (2014); 10.1063/1.4886355 State space realization of fractional order systems AIP Conf. Abstract. In this paper, we present a numerical study and a circuit design to prove existence of chaos in the fractional order Logistic delay system. In addition, we investigate an active control synchronization scheme in this system. Numerical and cicruit simulations show the effectiveness and feasibility of this method.
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