Abdon Atangana

Atangana went to primary and high school in a small village in Cameroon called Elig-Mfomo. He obtained his Honors, Master and PhD degrees in Applied Mathematics from the University of the Free State, where he now works as faculty, in 2010, 2011and 2013 respectively.[2] He was then offered the position of professor at the Institute for Groundwater Studies, University of the Free State, South Africa.

Atangana was included in the 2019 (Maths), 2020 (Cross-field) and the 2021 (Maths) Clarivate Web of Science lists of the World's top 1% scientists, one of only ten from South Africa in 2020, and he was awarded The World Academy of Sciences' (TWAS) inaugural Mohammed A. Hamdan award for contributions to science in developing countries.[3][4][5] In 2018 Atangana was elected a member of the African Academy of Sciences, and in 2021 a member of The World Academy of Sciences.[2][6] He also ranked number one in the world in mathematics, number 186 in the world in all the fields, and number one in Africa in all the fields, according to the Stanford list of 2% single-year table.[7]

Atangana has had some retractions of his publications, which he qualified unfair retractions.[8] Since then he has written about key steps that should be followed to reach a fair retraction.[9]

His research interests are methods and applications of partial and ordinary differential equations, fractional differential equations, perturbation methods, asymptotic methods, iterative methods, and groundwater modelling, fractal-fractional calculus, numerical methods.

The concept of fractional calculus has been recognized as powerful mathematical tools to replicate processes with nonlocal behaviors. In nature one can observed processes following power law behaviors. For example the distributions of a wide variety of physical, biological, and man-made phenomena approximately follow a power law over a wide range of magnitudes. There are processes that follow naturally decay behaviors, a clear example is the decay process of a human or animal flesh, the rates of certain types of chemical reactions depend on the concentration of one or another reactant. On the other hand, several real world problems display crossover behaviors from exponential decay law to power law, a clear example is the flow of groundwater from matrix soil to fractures formation.

To model processes with crossover behaviors from exponential decay to power law, In 2016, Atangana and Baleanu suggested differential operators based on the generalized Mittag-Leffler function. This function has several applications in real world, One of the applications of the Mittag-Leffler function is in modeling fractional order viscoelastic materials. Experimental investigations into the time-dependent relaxation behavior of viscoelastic materials are characterized by a very fast decrease of the stress at the beginning of the relaxation process and an extremely slow decay for large times. It can even take a long time before a constant asymptotic value is reached. Therefore, a lot of Maxwell elements are required to describe relaxation behavior with sufficient accuracy. This ends in a difficult optimization problem in order to identify a large number of material parameters. On the other hand, over the years, the concept of fractional derivatives has been introduced to the theory of viscoelasticity. Among these models, the fractional Zener model was found to be very effective to predict the dynamic nature of rubber-like materials with only a small number of material parameters. The solution of the corresponding constitutive equation leads to a relaxation function of the Mittag-Leffler type. It is defined by the power series with negative arguments. This function represents all essential properties of the relaxation process under the influence of an arbitrary and continuous signal with a jump at the origin.[10][11] Additionally to these properties of the generalized Mittag-Leffler function, Atangana and Baleanu aimed at introducing a fractional derivative with nonlocal and nonsingular kernel.

Their fractional differential operators are given below in Riemann-Liouville sense and Caputo sense respectively. For a function ${\displaystyle f(t)}$ of ${\displaystyle C^{1}}$ given by [12][13]

$${\displaystyle _{a}^{ABC}D_{t}^{\alpha }f(t)={\frac {AB(\alpha )}{1-\alpha }}\int _{a}^{t}f'(\tau )E_{\alpha }\left(-\alpha {\frac {(t-\tau )^{\alpha }}{1-\alpha }}\right)d\tau ,}$$

If the function is continuous, the Atangana-Baleanu derivative in Riemann-Liouville sense is given by:

$${\displaystyle _{a}^{ABC}D_{t}^{\alpha }f(t)={\frac {AB(\alpha )}{1-\alpha }}{\frac {d}{dt}}\int _{a}^{t}f(\tau )E_{\alpha }\left(-\alpha {\frac {(t-\tau )^{\alpha }}{1-\alpha }}\right)d\tau ,}$$

The kernel used in Atangana-Baleanu fractional derivative has some properties of a cumulative distribution function. For example, for all ${\displaystyle \alpha \in (0,1]}$, the function ${\displaystyle E_{\alpha }}$ is increasing on the real line, converges to ${\displaystyle 0}$ in ${\displaystyle -\infty }$, and ${\displaystyle E_{\alpha }(0)=1}$. Therefore, we have that, the function ${\displaystyle x\mapsto 1-E_{\alpha }(-x^{\alpha })}$ is the cumulative distribution function of a probability measure on the positive real numbers. The distribution is therefore defined, and any of its multiples, is called a Mittag-Leffler distribution of order ${\displaystyle \alpha }$. It is also very well-known that, all these probability distributions are absolutely continuous. In particular, the function Mittag-Leffler has a particular case ${\displaystyle E_{1}}$, which is the exponential function, the Mittag-Leffler distribution of order ${\displaystyle 1}$ is therefore an exponential distribution. However, for ${\displaystyle \alpha \in (0,1)}$, the Mittag-Leffler distributions are heavy-tailed. Their Laplace transform is given by:

${\displaystyle \mathbb {E} (e^{-\lambda X_{\alpha }})={\frac {1}{1+\lambda ^{\alpha }}},}$

This directly implies that, for ${\displaystyle \alpha \in (0,1)}$, the expectation is infinite. In addition, these distributions are geometric stable distributions.

To satisfy the fundamental theorem of fractional calculus with nonlocal and non-singular kernel, a fractional integral that is invertible with the Atangana-baleanu fractional derivative in Riemann-Liouville sense was obtained. The Atangana-Baleanu fractional integral of a continuous function is defined as.[12][13] The Atangana-Baleanu fractional integral of a continuous function is defined as:

$${\displaystyle ^{AB}{}_{a}I_{t}^{\alpha }f(t)={\frac {1-\alpha }{AB(\alpha )}}f(t)+{\frac {\alpha }{AB(\alpha )\Gamma (\alpha )}}\int _{a}^{t}\left(t-\tau \right)^{\alpha -1}f(\tau )\,d\tau }$$

In 2017 Atangana suggested the concept of fractal-fractional differential and integral operators (see Fractal derivative § Fractal-fractional calculus).[14] The concept was applied to model several real world problems.[15][14]

Global Differentiation is the procedure to calculate a derivative of a function f respect to another function g. The global derivative of a function y = f(x) respect to a function g(x) is a measure of the rate at which the value y of the function changes with respect to the change of the function g(x). It is called the global derivative of f with respect to g(x).

${\displaystyle m_{g}={\frac {{\text{change in }}f}{{\text{change in }}g}}={\frac {\Delta f(x)}{\Delta g(x)}}.}$

The function g(x) has some properties that are described in.[16]

The global derivative of the function f with respect to the function g(x) is defined as a limit:[16]

${\displaystyle D_{g}f(x)=\lim _{l\to x}{\frac {f(l)-f(x)}{g(l)-g(x)}}.}$

When the limit exists, f is said to be differentiable with respect to the function g(x). For example if ${\displaystyle g(x)=x^{\alpha }}$, ${\displaystyle \alpha >0}$ we have

${\displaystyle D_{g}f(x)=\lim _{l\to x}{\frac {f(l)-f(x)}{l^{\alpha }-x^{\alpha }}}.}$

Which is the fractal derivative. The concept of global derivative was extended within the framework of fractional calculus.[16] The corresponding integral of the above derivative is the well-known Riemann–Stieltjes integral.

Example of general derivative

${\displaystyle D_{x^{2}}x^{4}=\lim _{x\rightarrow x_{1}}{\frac {x^{4}-x_{1}^{4}}{x^{2}-x_{1}^{2}}}=\lim _{x\rightarrow x_{1}}x^{2}-x_{1}^{2}=2x_{1}^{2},}$

${\displaystyle D_{x^{4}}x^{2}=\lim _{x\rightarrow x_{1}}{\frac {x^{2}-x_{1}^{2}}{x^{4}-x_{1}^{4}}}=\lim _{x\rightarrow x_{1}}{\frac {1}{x^{2}+x_{1}^{2}}}={\frac {1}{2x_{1}^{2}}},}$

${\displaystyle D_{\cos x}\sin x=\lim _{x\rightarrow x_{1}}{\frac {\sin x-\sin x_{1}}{\cos x-\cos x_{1}}}=-{\frac {\cos x_{1}}{\sin x_{1}}},}$

${\displaystyle D_{\sin x}\cos x=\lim _{x\rightarrow x_{1}}{\frac {\cos x-\cos x_{1}}{\sin x-\sin x_{1}}}=-\tan x_{1}.}$

Convolution of global rate with kernels

Atangana extended this concept using the convolution approach. For example with the generalized Mittag-Leffler function, the following global fractional derivative can be obtained if :[16]

${\displaystyle K\left(t\right)={\frac {AB\left(\alpha \right)}{1-\alpha }}E_{\alpha }\left[-{\frac {\alpha }{1-\alpha }}t^{\alpha }\right]}$

The following global fractional derivatives are defined

${\displaystyle _{0}^{ABC}D_{g}^{\alpha }f\left(t\right)={\frac {AB\left(\alpha \right)}{1-\alpha }}\int _{0}^{t}D_{g}f\left(\tau \right)E_{\alpha }\left[-{\frac {\alpha }{1-\alpha }}\left(t-\tau \right)^{\alpha }\right]d\tau ,}$

${\displaystyle _{0}^{ABR}D_{g}^{\alpha }f\left(t\right)=={\frac {AB\left(\alpha \right)}{1-\alpha }}D_{g}\int _{0}^{t}f\left(\tau \right)E_{\alpha }\left[-{\frac {\alpha }{1-\alpha }}\left(t-\tau \right)^{\alpha }\right]d\tau .}$

Finally, using the differentiability of the function g(t) and letting

${\displaystyle {\frac {1}{g\prime (t)}}=\gamma \prime (t)}$,

we have the following expressions

${\displaystyle _{0}^{ABC}D_{g}^{\alpha }f\left(t\right)={\frac {AB\left(\alpha \right)}{1-\alpha }}\int _{0}^{t}{\frac {d}{d\tau }}f\left(\tau \right)\gamma \prime \left(\tau \right)E_{\alpha }\left[-{\frac {\alpha }{1-\alpha }}\left(t-\tau \right)^{\alpha }\right]d\tau ,}$

${\displaystyle _{0}^{ABR}D_{g}^{\alpha }f\left(t\right)={\frac {AB\left(\alpha \right)}{1-\alpha }}{\frac {d}{dt}}\int _{0}^{t}f\left(\tau \right)E_{\alpha }\left[-{\frac {\alpha }{1-\alpha }}\left(t-\tau \right)^{\alpha }\right]d\tau \gamma \prime \left(t\right).}$

It is worth noting that many other kernels can be used to recover other derivatives.[16]

Associated integral can be presented as follow:[16]

${\displaystyle u\left(t\right)={\frac {1}{\Gamma \left(1-\alpha \right)}}D_{g}\int _{0}^{t}f\left(\tau \right)\left(t-\tau \right)^{-\alpha }d\tau }$

Knowing that the integral is differentiable then:

${\displaystyle \left\{{\begin{array}{c}u\left(t\right)&=&{\frac {d}{dt}}{\frac {1}{\Gamma \left(1-\alpha \right)}}\int _{0}^{t}f\left(\tau \right)\left(t-\tau \right)^{-\alpha }d\tau {\frac {1}{g\prime \left(t\right)}}\\u\left(t\right)g\prime \left(t\right)&=&{\frac {d}{dt}}{\frac {1}{\Gamma \left(1-\alpha \right)}}\int _{0}^{t}f\left(\tau \right)\left(t-\tau \right)^{-\alpha }d\tau .\end{array}}\right.}$

Using the Riemann-Liouville

${\displaystyle f\left(t\right)={\frac {1}{\Gamma \left(\alpha \right)}}\int _{0}^{t}g\prime \left(\tau \right)u\left(\tau \right)\left(t-\tau \right)^{\alpha -1}d\tau .}$

So with power-law, we can define the following integral

${\displaystyle _{0}^{R}I_{g}^{\alpha }f\left(t\right)={\frac {1}{\Gamma \left(\alpha \right)}}\int _{0}^{t}g\prime \left(\tau \right)f\left(\tau \right)\left(t-\tau \right)^{\alpha -1}d\tau .}$

With exponential decay law, we have the following

${\displaystyle \left\{{\begin{array}{c}u\left(t\right)&=&D_{g}{\frac {M\left(\alpha \right)}{1-\alpha }}\int _{0}^{t}f\left(\tau \right)\exp \left(-{\frac {\alpha }{1-\alpha }}\left(t-\tau \right)\right)d\tau \\u\left(t\right)g\prime \left(t\right)&=&{\frac {d}{dt}}{\frac {M\left(\alpha \right)}{1-\alpha }}\int _{0}^{t}f\left(\tau \right)\exp \left(-{\frac {\alpha }{1-\alpha }}\left(t-\tau \right)\right)d\tau .\end{array}}\right.}$

Thus

${\displaystyle f\left(t\right)={\frac {1-\alpha }{M\left(\alpha \right)}}u\left(t\right)g\prime \left(t\right)+{\frac {\alpha }{M\left(\alpha \right)}}\int _{0}^{t}g\prime \left(\tau \right)u\left(\tau \right)d\tau .}$

Thus, the following integral

${\displaystyle _{0}^{CFR}I_{g}^{\alpha }f\left(t\right)={\frac {1-\alpha }{M\left(\alpha \right)}}f\left(t\right)g\prime \left(t\right)+{\frac {\alpha }{M\left(\alpha \right)}}\int _{0}^{t}g\prime \left(\tau \right)f\left(\tau \right)d\tau .}$

With Atangana-Baleanu derivative, we have the following

${\displaystyle \left\{{\begin{array}{c}u\left(t\right)&=&D_{g}{\frac {AB\left(\alpha \right)}{1-\alpha }}\int _{0}^{t}f\left(\tau \right)E_{\alpha }\left(-{\frac {\alpha }{1-\alpha }}\left(t-\tau \right)^{\alpha }\right)d\tau \\u\left(t\right)g\prime \left(t\right)&=&{\frac {AB\left(\alpha \right)}{1-\alpha }}{\frac {d}{dt}}\int _{0}^{t}f\left(\tau \right)E_{\alpha }\left(-{\frac {\alpha }{1-\alpha }}\left(t-\tau \right)^{\alpha }\right)d\tau .\end{array}}\right.}$

There using the Atangana-Baleanu integral, we have

${\displaystyle f\left(t\right)={\frac {1-\alpha }{AB\left(\alpha \right)}}u\left(t\right)g\prime \left(t\right)+{\frac {\alpha }{AB\left(\alpha \right)\Gamma \left(\alpha \right)}}\int _{0}^{t}g\prime \left(\tau \right)u\left(\tau \right)\left(t-\tau \right)^{\alpha -1}d\tau .}$

Thus the following integral can be defined

${\displaystyle \left\{{\begin{array}{c}_{0}^{ABR}I_{g}^{\alpha }f\left(t\right)&=&{\frac {1-\alpha }{AB\left(\alpha \right)}}f\left(t\right)g\prime \left(t\right)\\&&+{\frac {\alpha }{AB\left(\alpha \right)\Gamma \left(\alpha \right)}}\int _{0}^{t}g\prime \left(\tau \right)f\left(\tau \right)\left(t-\tau \right)^{\alpha -1}d\tau .\end{array}}\right.}$

A proper choice of ${\displaystyle g\left(t\right)}$ and the kernel helps to recover all existing integral operators whose derivatives are based on the concept of rate of change. If ${\displaystyle g\left(t\right)}$ is a weighted function also with an appropriate choice the of the kernel, one can recover all the weighted integrals.

In 2021, Atangana and Seda introduced the concept of piecewise differential and integral operators. They argued that many real world problems depict crossover behaviours, thus differential operators based on power law, exponential kernel and the generalized Mittag-Leffler function are unable to model such behaviours.[17] Some of examples of piecewise differential operators are given below Let f be differentiable, let g be a differentiable increasing non-zero function. The piecewise derivative is defined by

${\displaystyle _{0}^{PCCF}D_{g}^{\alpha }f{\text{ }}\left(t\right)=\left\{{\begin{array}{c}_{0}^{C}D_{g}^{\alpha }f{\text{ }}\left(t\right){\text{ if }}0\leq t\leq t_{0}\\_{t_{0}}^{CF}D_{g}^{\alpha }f{\text{ }}\left(t\right){\text{ if }}t_{0}\leq t\leq T\end{array}}\right.}$

where ${\displaystyle _{0}^{PCCF}D_{g}^{\alpha }}$ represents the global derivative with power-law on ${\displaystyle 0\leq t\leq t_{0}}$ and global derivative with exponential decay kernel on ${\displaystyle t_{0}\leq t\leq T.}$

A piecewise integral of f with respect to g is given as

${\displaystyle ^{P}J_{g}f{\text{ }}\left(t\right)=\left\{{\begin{array}{c}\int _{0}^{t}f{\text{ }}\left(\tau \right)g\prime \left(\tau \right)d\tau {\text{ if }}0\leq t\leq t_{1}\\{\frac {1}{\Gamma \left(\alpha \right)}}\int _{t_{1}}^{t}f{\text{ }}\left(\tau \right)\left(t-\tau \right)^{\alpha -1}d\tau {\text{ if }}t_{1}\leq t\leq T\end{array}}\right.}$

The piecewise derivative with global and exponential decay kernel are defined as:

${\displaystyle _{0}^{P}D_{g}^{\alpha }f{\text{ }}\left(t\right)=\left\{{\begin{array}{c}D_{g}f{\text{ }}\left(t\right){\text{ if }}0\leq t\leq t_{1}\\_{t_{1}}^{CF}D_{t}^{\alpha }f{\text{ }}\left(t\right){\text{ if }}t_{1}\leq t\leq T\end{array}}\right.}$

${\displaystyle _{0}^{P}D_{t}^{\alpha }f{\text{ }}\left(t\right)=\left\{{\begin{array}{c}D_{g}f{\text{ }}\left(t\right){\text{ if }}0\leq t\leq t_{1}\\_{t_{1}}^{CFR}D_{t}^{\alpha }f{\text{ }}\left(t\right){\text{ if }}t_{1}\leq t\leq T\end{array}}\right..}$

A piecewise integral of f with respect to g is given as

${\displaystyle ^{P}J_{g}f{\text{ }}\left(t\right)=\left\{{\begin{array}{c}\int _{0}^{t}f{\text{ }}\left(\tau \right)g\prime \left(\tau \right)d\tau {\text{ if }}0\leq t\leq t_{1}\\{\frac {1-\alpha }{M\left(\alpha \right)}}f{\text{ }}\left(t\right)+{\frac {\alpha }{M\left(\alpha \right)}}\int _{t_{1}}^{t}f{\text{ }}\left(\tau \right)d\tau {\text{ if }}t_{1}\leq t\leq T\end{array}}\right..}$

The piecewise derivative with global and Mittag-Leffler kernel is given as

${\displaystyle _{0}^{P}D_{t}^{\alpha }f{\text{ }}\left(t\right)=\left\{{\begin{array}{c}D_{g}f{\text{ }}\left(t\right){\text{ if }}0\leq t\leq t_{1}\\_{t_{1}}^{ABC}D_{t}^{\alpha }f{\text{ }}\left(t\right){\text{ if }}t_{1}\leq t\leq T\end{array}}\right..}$

A piecewise integral of f with respect to g is given as

${\displaystyle ^{P}J_{g}f{\text{ }}\left(t\right)=\left\{{\begin{array}{c}\int _{0}^{t}f{\text{ }}\left(\tau \right)g\prime \left(\tau \right)d\tau {\text{ if }}0\leq t\leq t_{1}\\{\frac {1-\alpha }{AB\left(\alpha \right)}}f{\text{ }}\left(t\right)+{\frac {\alpha }{AB\left(\alpha \right)\Gamma \left(\alpha \right)}}\int _{t_{1}}^{t}f{\text{ }}\left(\tau \right)\left(t-\tau \right)^{\alpha -1}d\tau {\text{ if }}t_{1}\leq t\leq T\end{array}}\right..}$

In 2019, Atangana and Mekkaoui Toufik suggested a complex number with two imaginary parts. They are argued that Humans being live in three-dimensional space therefore, they can only accurately visualize processes taking place in one, two and three dimensions. While the set of bi-complex numbers and quaternion have been in fashion as they have attracted attention of many researchers in physics and related branches, they do not really represent processes taking place in the space where human being are located.[18]

A trinition is defined as a set T, such that ${\displaystyle z\in T}$ if:

${\displaystyle z=a+bi+cj,(a,b,c)\in R^{3}}$

where iand j are fundamental Trinition units. As many other sets, the following basic properties can be verified

Addition Let ${\displaystyle z_{1},z_{2}\in T}$ , then

${\displaystyle z_{1}+z_{2}=a+bi+cj+a_{1}+ib_{1}+jc_{1}=(a+a_{1})+(b+b_{1})i+j(c+c_{1})=z_{2}+z_{1}.}$

Multiplication in Trinition : The motivation was based on the fact that, the trinition is the corresponding complex set of ${\displaystyle R^{3}}$, the same as with C and ${\displaystyle R^{2}}$ .

${\displaystyle i^{2}=j^{2}=-1,i.j=1,j.i=-1.}$

The above can be used to generate the following iteration that could lead obviously to fractal mapping like the Julia set for complex number.

${\displaystyle z_{n+1}=z_{n}^{2}+c.}$

From the above the following corresponding Julia set is obtained

${\displaystyle \left\{{\begin{array}{c}x_{n+1}=x_{n}^{2}-y_{n}^{2}+z_{n}^{2}+c_{x}\\y_{n+1}=2x_{n}y_{n}+c_{y}\\z_{n+1}=2x_{n}z_{n}+c_{z}.\end{array}}\right.}$

An interesting mapping can be obtained and called African mystical bird mapping was presented as:

${\displaystyle \left\{{\begin{array}{c}x_{n+1}=y_{n}+\alpha \left(1-0.005y_{n}^{2}\right)y_{n}+\mu x_{n}+{\frac {2\left(1-\mu \right)x_{n}^{2}}{1+x_{n}^{2}}}\\y_{n+1}=-x_{n}+\mu x_{n+1}+{\frac {2\left(1-\mu \right)x_{n+1}^{2}}{1+x_{n+1}^{2}}}\\z_{n+1}=-\lambda y_{n}+\varepsilon y_{n+1}+{\frac {2\left(1-\mu \right)y_{n+1}^{2}}{1+y_{n+1}^{2}}}\end{array}}\right.}$

A partial conjugate is defined as

${\displaystyle z^{-i}=a-ib+jc,}$.

${\displaystyle \ z^{-j}=a+ib-jc.}$.

The total conjugate is

${\displaystyle {\overline {z}}=a-ib-jc.}$

${\displaystyle \forall z\in T}$ there exists two angles ${\displaystyle \alpha ,\beta }$ such that "(a,b)" and "c" are related. Then we define

${\displaystyle \left\{{\begin{array}{c}trisinus\left(\alpha ,\beta \right)&=&{\frac {b}{\sqrt {a^{2}+b^{2}+c^{2}}}}\\triconus\left(\alpha ,\beta \right)&=&{\frac {a}{\sqrt {a^{2}+b^{2}+c^{2}}}}\\trinus\left(\alpha ,\beta \right)&=&{\frac {c}{\sqrt {a^{2}+b^{2}+c^{2}}}}\end{array}}\right.}$

where

${\displaystyle \beta =\arctan \left({\frac {y}{x}}\right),}$

${\displaystyle \alpha =\arccos \left({\frac {x}{r}}\right)}$

Here

${\displaystyle \left\{{\begin{array}{c}triconus\left(\alpha ,\beta \right)&=&\sin \alpha \cos \beta \\trisinus\left(\alpha ,\beta \right)&=&\sin \alpha \sin \beta \\trinus\left(\alpha ,\beta \right)&=&\cos \alpha .\end{array}}\right.}$

We present the following relation

${\displaystyle \left(triconus\left(\alpha ,\beta \right)\right)^{2}+\left(trisinus\left(\alpha ,\beta \right)\right)^{2}+\left(trinus\left(\alpha ,\beta \right)\right)^{2}=1.}$

The above is Pythagoras theorem in the Trinition.