The Limited Validity of the Fractional Euler Finite Difference Method
and an Alternative Definition of the Caputo Fractional Derivative to
Justify Modification of the Method
DOMINIC P. CLEMENCE-MKHOPE, ZACHARY DENTON
Department of Mathematics and Statistics,
North Carolina A&T State University,
1601 East Market Street, Greensboro, NC 27411,
USA
Abstract: - A method, advanced as the fractional Euler finite difference method (FEFDM), a general method for
the finite difference discretization of fractional initial value problems (IVPs) for for the Caputo
derivative, is shown to be valid only for . This is accomplished by establishing, through a recently
proposed generalized difference quotient representation of the fractional derivative, that the FEFDM is valid
only if a property of the Mittag-Leffler function holds that has only been shown to be valid only for . It is
also shown that the FEFDM is inconsistent with the exact discretization of the IVP for the Caputo fractional
relaxation equation. The generalized derivative representation is also used to derive a modified generalized
Euler’s method, its nonstandard finite difference alternative, their improved Euler versions, and to recover a
recent result by Mainardi relating the Caputo and conformable derivatives.
Key-Words: - Caputo fractional derivative, fractional Euler finite difference method (FEFDM), modified
FEFDM, fractional initial value problem, fractional relaxation equation
1 Introduction
The Caputo fractional derivative (FD) is defined,
[1], as
(1)
for , with which this paper is concerned.
It is one of the classical fractional derivatives, [2],
upon which most theory on the subject has been
developed by various authors and it has been widely
applied across many areas of science and
engineering, [3], and references therein for
example).
Since most fractional differential equations
systems do not have exact analytic solutions,
numerical approximation methods must be
developed. Among the many numerical methods,
[4], [5], [6], [7], [8], [9], that have been proposed to
solve initial value problems (IVPs) for differential
equations with the Caputo FD,
(2)
the finite difference method
,
where
, (3)
was proposed in, [5], which has been widely cited.
Method (3), which we will refer to as the fractional
Euler finite difference method (FEFDM) (and is
often referred to as the generalized Euler method),
has been used in direct applications, [10], [11], [12],
[13], as well as in developing other methods, [14],
[15], [16], [17], [18], for the discretization of the
fractional IVP (2). It is justified by applying
Taylor’s expansion for the Caputo FD, which is
proposed in, [19], and is used to develop an
algorithm (4) below for solving the IVP (2):
+
,
(4)
which will be referred to as the Odibat-Momani
finite difference method (OMFDM).
The purpose of this article is to show that the
FEFDM, method (3), as a model for (2) is valid only
Received: June 29, 2023. Revised: September 26, 2023. Accepted: October 19, 2023. Published: November 9, 2023.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.91
Dominic P. Clemence-Mkhope, Zachary Denton
E-ISSN: 2224-2880
831
Volume 22, 2023
for . Therefore, any algorithm based on it,
such as the OMFM (4) wherein the FEFDM (3) is
used as a predictor and a modified trapezoidal rule
as corrector, is consequently valid only at . A
modified fractional Euler finite difference method
(MFEFDM
,
(5)
as well as its improved Euler counterpart, are
proposed, where and are respectively
the one- and two-parameter Mittag-Leffler (ML)
functions. This modification is accomplished using
the expression (6) below, recently proposed in, [20],
of the Caputo FD in terms of the ML function:
,
where
(6)
The rest of this article is organized as follows. In
the next section, the ESDDFD method for difference
quotient representations for non-integer derivatives,
generalized fractional derivatives, and subsequent
Euler method extensions are recalled. Section 3
recalls the derivation of the FEM from, [1], and it is
shown that it is valid only for . In Section 4,
the ESDDFD method difference quotient
representations and generalized fractional
derivatives for the Caputo FD, and subsequent Euler
method extensions are presented. Numerical
experiments are presented in Section 5 assessing the
accuracy, against analytic solutions, of the FEFDM
and MFEFDM for two examples. A discussion in
Section 6 of the theoretical and experimental results
presented, as well as recommendations based on
those results, concludes the article.
2 ESDDFD Difference Quotient
Representation
The exact spectral derivative discretization finite
difference (ESDDFD) method was introduced in,
[20], in which it is generally assumed that the IVP
(2) is being discretized on intervals of the form
and the following were presented.
Definition 2.1. For a given definition of an FD, let
denote the analytic solution of IVP for
the fractional relaxation equation (FRE):
. (7)
Then a corresponding difference quotient
representation (DQR) of the Caputo type consistent
with that derivative is
Taking the limit as in the equation above
yields the following alternative definition of the
derivative associated with :
Definition 2.2. Given a real-valued function on
, the generalized fractional derivative (GFD)
associated with has the following
alternative definition:
,
where
is understood to mean
.
The identifications
,
(8)
applied in Definitions 2.1 and 2.2 yields the
following discretization rule for
as a
corollary.
Corollary 2.1. Let be as in Definitions
2.1. Then the following are consistent discrete Euler
and nonstandard finite difference (NSFD)
representations of
:
Generalized Fractional Euler:
,
Generalized Fractional NSFD:
,
where is defined as follows:
The denominator function in Corollary
2.1 is a complex function of both the step size
and lattice point , and is described in
[20], as a fractional generalization of the
nonstandard finite difference (NSFD) denominator,
[21].
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.91
Dominic P. Clemence-Mkhope, Zachary Denton
E-ISSN: 2224-2880
832
Volume 22, 2023
Clear corollaries to the foregoing are the
following Euler discretization rules for the IVP (2),
which provide justified extensions of the Euler
method to the GFD:
Corollary 2.2. The following discrete
representations are fractional generalizations of the
Euler finite difference method for the GFD valid for
:
Generalized FEFDM:
Generalized Fractional NSFD Method:
The following alternate definition of the
conformable fractional derivative (CFD), [22],
given in, [23], will be used to arrive at the
MFEFDM (5). It can be obtained by setting
, in Definitions 2.1 and 2.2
above.
Definition 2.3. Given a real-valued function on
, the conformable fractional derivative has the
following alternative definition:
where
is understood to mean
.
The following, which gives the relationship
between the integer derivative and NIDs, will also
be used to arrive at the MFEFDM, given by (5).
Proposition 2.1 If and are both
first-order differentiable, then the following also
holds:
Proof
The proof follows directly from Definitions 2.1 and
2.2:
3 The Generalized Fractional Euler
Finite Difference Method (FEFDM)
In this section, justification of the generalized
fractional Euler finite difference method is recalled,
and proof is presented of its limited validity.
3.1 Justification of the FEFDM
The FEFDM, given by (3), for the IVP (2), is
obtained in, [5], by considering a Caputo FD power
series expansion, [19], as follows. It is assumed that
for each there exists so that the following is
true:
. (9)
Letting and
substituting
into (9) results
in
,
or, equivalently
.
(10)
For small enough, ignoring the second term
on the right-hand side of (10) yields the fractional
Euler finite difference method (FEFDM) given in
Eqn. (3):
,
which reduces to the usual Euler’s method for
.
3.2 Limited Validity of the FEFDM
Next, the following is proved:
Proposition 3.1 The FEFDM as given by Eqn. (3) is
valid only for .
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.91
Dominic P. Clemence-Mkhope, Zachary Denton
E-ISSN: 2224-2880
833
Volume 22, 2023
Proof
Substituting , the solution of
the FRE for the Caputo FD, into Definition 2.2
results in the generalized representation:
. (11)
For small enough, therefore, and making the
identifications (8) in Eqn. (11), or using Corollary
2.1, results in the following discrete representation
of the Caputo FD,
:
.
Now, if the FEFDM is valid, then we have the
following two equivalent discrete representations of
the IVP (7) with Caputo FD
:
and
. (12)
Therefore, from the equivalence of the left-hand
sides in (12) above we conclude the following:
. (13)
However, the only way that Eqn. (13) holds that the
following identity holds,
,
(14)
so that there follows
.
Since the left-hand side of the first representation
in (12) is an exact discretization of the RE for the
Caputo FD for , it is consistent with the
representation of the Caputo FD for .
Since the identity (14) has been shown by example,
[24], to not hold for , we conclude therefore
that (13) also holds only for . From the
preceding statements, we conclude therefore that the
right-hand side (RHS) of (13) is consistent with the
representation of the Caputo FD only for , and
hence that the FEFDM, since it derives from the
RHS of (13), is valid only for .
4 Alternative Definition of the
Caputo FD and Justification of the
MFEM
In this section, an alternative definition of the
Caputo derivative is presented and used to derive a
modified fractional Euler finite difference method
(MFEFDM).
4.1 Alternative Definition of the Caputo FD
The derivation of a modified FEFDM is based on
the exact discretization of the initial value problem
for the FRE, obtained from using the solution of the
FRE for the Caputo FD, in
Definitions 2.1 and 2.2, and leads to the following
DQR and GFD for the Caputo FD:
Definition 4.1. The Caputo fractional derivative has
the following difference quotient representation
(DQR):
, where
and associated generalized Caputo derivative as
given by (6):
.
The following result about the basic properties of
is a particular case of Theorem 2.1.6 of, [20].
Theorem 4.1. Let and the functions
be -differentiable at a point . Then, for
all real-valued constants , the following
properties hold:
(1).
(2).
(3).
(4).
(5).
(6). If is first-order differentiable, then the
following also holds:
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.91
Dominic P. Clemence-Mkhope, Zachary Denton
E-ISSN: 2224-2880
834
Volume 22, 2023
Proof (of (6))
The proofs are elementary proofs that are omitted
here: 1,2,3,5 follow directly from Definition 3.1
while 4),6) follow directly from Proposition 2.1,
with and use of the identity
Remark: The formula in Theorem 4.1 (6) is the
same as that recently obtained by, [25], from
consideration of the relationship between relaxation
equations of integer order and those of fractional
order. The ESDDFD therefore offers, through
Proposition 2.1, a generalization of the Mainardi
result as well as the relationship between the Caputo
FD and the integer derivative.
From Definition 4.1, we have the following:
Corollary 4.2. The Caputo derivative has the
following generalized fractional derivative (GFD):
,
where is defined as follows:
.
Proof
The proof follows directly from Theorem 4.1 (6),
noting that
,
where
denotes the CFD, and the use of the
alternate definition of the CFD given in Definition
2.3.
4.2 Modification of the FEFDM and Its
NSFD Alternative
The proposed modification of the FEFDM and its
alternative follow directly from the GFD in
Corollary 4.2 and the DQR in Definition 4.1, with
the identifications in Eqn. (8), which yields the
following possible discretizations for the Caputo
FD:
Caputo FEFDM:
Caputo FNSFD:
Since the OMFDM may be viewed as a two-step
improved version of the FEFDM, an improved
version of the MFEFDM may be constructed for
comparison. A corollary to the foregoing, and given
Corollary 2.2, are the following Euler discretization
rules for the IVP (2) for the Caputo FD, which
justifies the modified FEFDM (MFEFDM) and its
NSFD alternative as extensions of the Euler method
to the Caputo FD:
Corollary 4.3. The following discrete
representations are generalizations of the (forward)
Euler method and its two-step improved versions for
the Caputo FD valid for
Modified FEFDM (MFEFDM):
Improved Modified FEFDM (IMFEFDM):
Fractional NSFD (FNSFD):
Improved Fractional NSFD (IFNSFD)::
5 Numerical Experiments
To further demonstrate that the FEFDM is not a
viable extension of the Euler method to the Caputo
FD for and to validate the suggested
alternatives, two examples are presented.
Example 1
.
Example 1 is used to justify the FEFDM in, [5],
and to validate the OMFDM in, [26], and those
results are in agreement with those of, [27]. The
OMFM model for Example 1,
, (15)
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.91
Dominic P. Clemence-Mkhope, Zachary Denton
E-ISSN: 2224-2880
835
Volume 22, 2023
is compared to the, respectively, FEFDM,
MFEFDM, IMFEFDM, FNSFD, and IFNSFD
models:
(16)
(17)
(18)
where
is of Eqn. (17);
(19)
,
where
is of Eqn. (19)
(20)
With the OMFDM (15) viewed as an improved
version of the FEFDM (16), comparisons against the
analytic solutions are presented in Figure 1 and
Figure 2 below using discrete representations
obtained from the FEFDM, MFEFDM (17), and
FNSFD (19), as well as their improved versions
(respectively, OMFDM, IMFEFDM (18), and
IFNSFD (20))
a b
c d e
f g i
Fig. 1: Solution profiles for Example 1 when (a. FEFDM, b. OMFDM; (c–e) MFEFDM and
IFEFDM (c. h=0.1, d. h=0.01, e. h=0.001); (f–i) NSFD and INSFD (f. h=0.1, g. h=0.01, i. h=0.001)
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.91
Dominic P. Clemence-Mkhope, Zachary Denton
E-ISSN: 2224-2880
836
Volume 22, 2023
a b
c d e
f g i
Fig. 2: Solution profiles for Example 1 when . (a. FEFDM, b. OMFDM; (c–e) MFEFDM and
IFEFDM (c. h=0.1, d. h=0.01, e. h=0.001); (f–i) NSFD and INSFD (f. h=0.1, g. h=0.01, i. h=0.001)
It is clear from Figure 1 and Figure 2 that the
FEFDM does not perform well for all step sizes; this
inferior performance persists for all values of
. While it performs better than the FEFDM, the
OMFDM is seen to under-perform both the
MFEFDM and the FNSFD as well as their improved
versions for all step sizes. The absolute and
percentage errors for and various step sizes
are presented in Table 1 below to further quantify
these performance differences for all the six
considered method.
Table 1. Error-values for Example 1 when
h
error
FEFDM
OMFM
MFEFDM
IMFEFDM
FNSFD
IFNSFD
0.1
Abs E
0.415469
0.054394
0.060024
0.002517
0
0.038582
% E
97.16684
12.72124
14.03785
0.588624
0
9.023249
0.01
Abs E
0.427577
0.011239
0.007778
0.000703
0
0.006339
% E
99.99852
2.62848
1.819015
0.164483
0
1.482554
0.001
Abs E
0.427584
0.003192
0.000998
9.15E-05
0
0.00088
% E
100
0.746423
0.233314
0.021411
0
0.205787
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.91
Dominic P. Clemence-Mkhope, Zachary Denton
E-ISSN: 2224-2880
837
Volume 22, 2023
Example 2
.
Example 2 is a slight extension of Example 1.
Comparisons against the analytic solutions for
Example 2 are presented in graphical form in Figure
3 and Figure 4 below using discrete representations
obtained from the Caputo fractional Euler, modified
fractional Euler, and ESDDFD-based NSFD Euler
methods (respectively, FEFDM, MFEFDM, and
FNSFD), and their improved versions (respectively,
OMFM, IMFEFDM, and IFNSFD).
a. b
c d e
f g i
Fig. 3: Solution profiles for Example 2 when . (a. FEFDM, b. OMFDM; (c–e) MFEFDM and IFEFDM
(c. h=0.1, d. h=0.01, e. h=0.001); (f–i) NSFD and INSFD (f. h=0.1, g. h=0.01, i. h=0.001)
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.91
Dominic P. Clemence-Mkhope, Zachary Denton
E-ISSN: 2224-2880
838
Volume 22, 2023
Consistent with the results of Example 1, it is
seen from Figure 3 above and Figure 4 below that
the FEFDM under-performs all the other methods
while the MFEFDM, the FNSFD, and their
improved versions all outperform the OMFDM.
The absolute and percentage errors from
Example 2 for and various step sizes are
presented in Table 2 below to further quantify the
performance differences for all the six considered
methods.
a. b.
c d e
f g i
Fig. 4: Solution profiles for Example 2 when . (a. FEFDM, b. OMFDM; (c–e) MFEFDM and
IFEFDM (c. h=0.1, d. h=0.01, e. h=0.001); (f–i) NSFD and INSFD (f. h=0.1, g. h=0.01, i. h=0.001)
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.91
Dominic P. Clemence-Mkhope, Zachary Denton
E-ISSN: 2224-2880
839
Volume 22, 2023
Table 2. Error-values for Example 2 when
h
error
FEFDM
OMFM
MFEFDM
IMFEFDM
FNSFD
IFNSFD
0.1
Abs E
0.415469
0.132849
0.060024
0.002517
2.11E-15
0.038582
% E
72.58168
23.20846
10.486
0.439691
3.66E-13
6.740186
0.01
Abs E
0.427577
0.041251
0.007778
0.000703
2.11E-15
0.006339
% E
74.69689
7.206459
1.358768
0.122865
3.66E-13
1.107438
0.001
Abs E
0.427584
0.013144
0.000998
9.15E-05
1.55E-15
0.00088
% E
74.69799
2.296192
0.174281
0.015993
2.66E-13
0.153718
6 Conclusion
A discretization method for the fractional initial
value problem, with the Caputo fractional
derivative, has been considered that extends the
integer Euler method and is termed the generalized
fractional Euler’s method in recent literature; its
justification using a fractional series expansion is
recalled. It has been shown that the method is valid
only for . A modified generalized fractional
Euler’s method and a corresponding nonstandard
method are proposed along with their improved
Euler counterparts. Numerical experiments are
presented comparing the FEFDM with the Odibat-
Momani algorithm derived from the FEFDM and
the four suggested alternative fractional Euler and
improved fractional Euler methods. Graphical
evidence and tabulation of absolute and percentage
errors show that the FEFDM has very large errors
and that the proposed methods outperform both the
FEFDM and the Odibat-Momani algorithm. The
proposed methods have the potential to improve the
numerical simulation of models of the form (2) in
direct applications (such as in, [10], [11], [12],
[13]), as well as in developing other methods (such
as in, [14], [15], [16], [17], [18]). As a next step, the
authors intend to apply these methods to the
numerical simulation and analysis of various
fractional disease models.
References:
[1] I. Podlubny, Fractional Differential
Equations, Academic Press, 1999.
[2] M. Dalir and M. Bashour, Applications of
Fractional Calculus Applied Mathematical
Sciences, 4(21), 2010, pp.1021–1032
[3] M. Caputo, Linear Models of Dissipation
whose Q is almost Frequency Independent-II.
Geophys. J. R. ustr. SOC., 13, 1967, pp.529–
539
[4] K. Diethelm, N.J. Ford, A.D. Freed, and Yu.
Luchko, Algorithms for the fractional
calculus: a selection of numerical methods.
Comput. Meth. Appl. Mech. Engrg., 194,
2005, pp.743–773.
[5] Z. M.Odibat and S. Momani, An Algorithm
for the numerical solution of differential
equations of fractional order, Journal of
applied mathematics & informatics, 26 (12),
2008, pp.15–27
[6] C. Li and F. Zeng, Finite Difference Methods
For Fractional Differential Equations.
International Journal of Bifurcation and
Chaos, 22(4), 2012, 1230014 (28 pages)
[7] D. Baleanu, K. Diethelm, E. Scalas, and J.J.
Trujillo, Fractional Calculus: Models and
numerical methods, World Scientific, 2012.
[8] G-h.Gao, Z-z. Sun, and H-w. Zhang, A new
fractional numerical differentiation formula to
approximate the Caputo fractional derivative
and its applications, Journal of Computational
Physics, 259, 2014, pp.33–50
[9] R. Garrappa, Numerical Solution of Fractional
Differential Equations: A Survey and a
Software Tutorial. Mathematics. 6(2),
2018:16.
https://doi.org/10.3390/math6020016
[10] M.M. Khadera and M. Adel, Numerical
treatment of the fractional SIRC model and
influenza A using generalized Euler method,
J. of Modern Methods in Numerical Math 6:1,
2015, pp.44–56
[11] D. Yaro, S.K. Omari-Sasu, P. Harvim, A.W.
Saviour, and B.A. Obeng, Generalized Euler
Method for Modeling Measles with Fractional
Differential Equations, International Journal
of Innovative Research & Development, 4(4),
2015, pp.358–366
[12] D. Vivek, K. Kanagarajan, and S.
Harikrishnan, Numerical solution of
fractional-order logistic equations by
fractional Euler’s method, International
Journal for Research in Applied Science &
Engineering Technology (IJRASET), 2016,
4(III), pp.775–780
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.91
Dominic P. Clemence-Mkhope, Zachary Denton
E-ISSN: 2224-2880
840
Volume 22, 2023
[13] Y.Y.Y. Noupoue, Y. Tandoğdu, and M.
Awadalla, On numerical techniques for
solving the fractional logistic differential
equation, Advances in Difference Equations,
108 (2019), 2019, pp.1–13.
https://doi.org/10.1186/s13662-019-2055-y
[14] C. Li and F. Zeng, The Finite Difference
Methods for Fractional Ordinary Differential
Equations, Numerical Functional Analysis
and Optimization, 34(2), 2013, pp.149–179,
DOI: 10.1080/01630563.2012.706673:
https://doi.org/10.1080/01630563.2012.70667
3
[15] N.A. Khan, O.A. Razzaq, and F. Riaz,
Numerical Simulations For Solving Fuzzy
Fractional Differential Equations By Max-
Min Improved Euler Methods, JACSM, 7(1),
2015, pp.53–83
[16] F.S. Fadhel and H.H. Khayoon, Numerical
Solution of Ordinary Differential Equations of
Fractional Order Using Variable Step Size
Method, Special Issue: 1st Scientific
International Conference, College of Science,
Al-Nahrain University, 21-22/11/2017, Part I,
pp.143–149, [Online],
https://www.iasj.net/iasj/download/875d174d
6dd06236 (Accessed Date: October 17, 2023)
[17] H.F. Ahmed, Fractional Euler Method; An
Effective Tool For Solving Fractional
Differential Equations, Journal of the
Egyptian Mathematical Society, 26(1), 2018,
pp.38–43. DOI: 10.21608/JOEMS.2018.9460
[18] M.S. Mechee and S.H. Aidib, Generalized
Euler and Runge-Kutta methods for solving
classes of fractional ordinary differential
equations, Int. J. Nonlinear Anal. Appl. 13(1),
2022, pp.1737–1745
http://dx.doi.org/10.22075/ijnaa.2022.5788
[19] Z. Odibat and N. Shawagfeh, Generalized
Taylor’s formula. Mathematics and
Computation, 186(1), 2007, 286–293.
DOI: 10.1016/j.amc.2006.07.102
[20] D.P. Clemence-Mkhope, Spectral Non-integer
Derivative Representations and the Exact
Spectral Derivative Discretization Finite
Difference Method for the Fokker-Planck
Equation. 2021,
[Online], https://ui.adsabs.harvard.edu/link_ga
teway/2021arXiv210602586C/arxiv:2106.025
86 (Accessed Date: October 17, 2023)
[21] R.E. Mickens, Nonstandard Finite Difference
Schemes: Methodology And Applications,
World Scientific Publishing Company, 2020.
[22] R. Khalil, M. Al Horani, A. Yousef, and M.
Sababheh. A new definition of fractional
derivative. J. Comput. Appl. Math. 2014, 264,
pp.65–70.
[23] D.P. Clemence-Mkhope and B.G.B.
Clemence-Mkhope, The Limited Validity of
the Conformable Euler Finite Difference
Method and an Alternate Definition of the
Conformable Fractional Derivative to Justify
Modification of the Method. Mathematical
and Computational Applications. 2021;
26(4):66.
https://doi.org/10.3390/mca26040066
[24] J. Peng and K. Li, A note on property of the
Mittag-Leffler function. Journal of
Mathematical Analysis and Applications,
370(2), 2010, pp.635–638
[25] F. Mainardi, A Note on the Equivalence of
Fractional Relaxation Equations to
Differential Equations with Varying
Coefficients. Reprinted from: Mathematics
2018, 6,8, DOI: 10.3390/math6010008
[26] S.S. Zeid, M. Yousefi, and A.V. Kamyad,
Approximate Solutions for a Class of
Fractional-Order Model of HIV Infection via
Linear Programming Problem. American
Journal of Computational Mathematics, 6,
2016, pp.141–152.
DOI: 10.4236/ajcm.2016.62015.
[27] Y. Ding and H. Ye, A fractional-order
differential equation model of HIV infection of
CD4+ T-cells, Mathematical and Computer
Modelling, 50 (3–4), 2009, pp.386–392,
https://doi.org/10.1016/j.mcm.2009.04.019.
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed to the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.91
Dominic P. Clemence-Mkhope, Zachary Denton
E-ISSN: 2224-2880
841
Volume 22, 2023
