Muhammad Altaf Khan completed his master’s degree in Applied mathematics from the Kohat University of Science in Technology in 2008. He completed his MS degree from Islamia college University Peshawar in 2012 in Applied mathematics and his research area is Mathematical Biology. He did his Ph.D. in Applied mathematics from Abdul Wali Khan University Mardan in 2015. He served as Lecturer in Abdul Wali Khan Mardan in the Department of Mathematics from 2015-2016 under the Umbrella of faculty development program. After that, he joined the City University of Science and Information Technology Peshawar and served there as Assistant Professor for the period August 2017-September 2019. He joined the post-doctorate position on October 01, 2019-December 2021 at the University of the Free State of South Africa. He also worked as a Researcher at Ton Duc Thang University Vietnam, March 2020-February 2021. He has been an Adjunct Professor at Airlangga University Indonesia for the period March 2021-Till February 2022. He is currently a Research Fellow at the University of the Free State of South Africa. He organized many special issues in reputable ISI journals, such as Advances in Difference equations, Results in Physics, Fractals, Alexandria Engineering Journals, Symmetry MDPI, etc. He is the author of more than 250 publications. He is an active editor in some reputable journals. He is an active researcher and worked working in fractional differential equations, evolutionary differential equations, optimal control problems, infectious diseases modeling, social sciences problems modeling, etc. He is a reviewer for many international journals of Springer, Wily, Hindawi, Elsevier, etc. Abdon Atangana works at the Institute for Groundwater Studies, University of the Free State, Bloemfontein, South Africa as a Full Professor. His research interests are, but are not limited to, fractional calculus and applications, numerical and analytical methods, and modeling. He is the author of more than 270 research papers and five books in top-tier journals of applied mathematics and groundwater modeling. He is the founder of numerous mathematical operators, for example, fractional differential and integral operators with singular and nonlocal kernels (Atangana-Baleanu derivatives and integral), fractal-fractional calculus, and piecewise calculus, and some numerical methods. He serves as an editor in top-tier journals in various fields of study. He has been invited as a plenary, and keynote speaker at more than 30 international conferences. He was elected Highly Cited Mathematician in 2019, Highly Cited Mathematician with Crossfield Impact in 2020, and Highly Cited Mathematician in 2021. He is a recipient of the Obada price 2020, the TWAS-Hamdan Award 2020, and many others. He is a fellow of the World Academia of Science.

• 1. Basic principle of nonlocalities. 2. Basic of fractional operators. 3. Definitions of fractal-fractional operators with numerical approximations. 4. Error analysis. 5. Existence and uniqueness of fractal fractional differential equations. 6. A numerical solution of fractal-fractional ODE with linear interpolation. 7. Numerical scheme of fractal-fractional ODE with middle point interpolation. 8. Fractal-Fractional Euler method. 9. Application of fractal-fractional operators to a chaotic model. 10. Fractal-fractional Modified Chua chaotic attractor. 11. Application of fractal-fractional operators to study a new chaotic model. 12. Fractal-fractional operators and their application to a chaotic system with sinusoidal component. 13. Application of fractal-fractional operators to four-scroll chaotic system. 14. Application of fractal-fractional operators to a novel chaotic model. 15. A 4D chaotic system under fractal-fractional operators. 16. Self-excited and hidden attractors through fractal-fractional operators. 17. Dynamical analysis of a chaotic model in fractal-fractional operators. 18. A chaotic cancer model in fractal-fractional operators. 19. A multiple chaotic attractor model under fractal-fractional operators. 20. The dynamics of multiple chaotic attractor with fractal-fractional operators. 21. Dynamics of 3D chaotic systems with fractal-fractional operators. 22. The hidden attractors model with fractal-fractional operators. 23. An SIR epidemic model with Fractal-fractional derivative. 24. Application of fractal-fractional operators to COVID-19 infection.

Muhammad Altaf Khan completed his master’s degree in Applied mathematics from the Kohat University of Science in Technology in 2008. He completed his MS degree from Islamia college University Peshawar in 2012 in Applied mathematics and his research area is Mathematical Biology. He did his Ph.D. in Applied mathematics from Abdul Wali Khan University Mardan in 2015. He served as Lecturer in Abdul Wali Khan Mardan in the Department of Mathematics from 2015-2016 under the Umbrella of faculty development program. After that, he joined the City University of Science and Information Technology Peshawar and served there as Assistant Professor for the period August 2017-September 2019. He joined the post-doctorate position on October 01, 2019-December 2021 at the University of the Free State of South Africa. He also worked as a Researcher at Ton Duc Thang University Vietnam, March 2020-February 2021. He has been an Adjunct Professor at Airlangga University Indonesia for the period March 2021-Till February 2022. He is currently a Research Fellow at the University of the Free State of South Africa. He organized many special issues in reputable ISI journals, such as Advances in Difference equations, Results in Physics, Fractals, Alexandria Engineering Journals, Symmetry MDPI, etc. He is the author of more than 250 publications. He is an active editor in some reputable journals. He is an active researcher and worked working in fractional differential equations, evolutionary differential equations, optimal control problems, infectious diseases modeling, social sciences problems modeling, etc. He is a reviewer for many international journals of Springer, Wily, Hindawi, Elsevier, etc. Abdon Atangana works at the Institute for Groundwater Studies, University of the Free State, Bloemfontein, South Africa as a Full Professor. His research interests are, but are not limited to, fractional calculus and applications, numerical and analytical methods, and modeling. He is the author of more than 270 research papers and five books in top-tier journals of applied mathematics and groundwater modeling. He is the founder of numerous mathematical operators, for example, fractional differential and integral operators with singular and nonlocal kernels (Atangana-Baleanu derivatives and integral), fractal-fractional calculus, and piecewise calculus, and some numerical methods. He serves as an editor in top-tier journals in various fields of study. He has been invited as a plenary, and keynote speaker at more than 30 international conferences. He was elected Highly Cited Mathematician in 2019, Highly Cited Mathematician with Crossfield Impact in 2020, and Highly Cited Mathematician in 2021. He is a recipient of the Obada price 2020, the TWAS-Hamdan Award 2020, and many others. He is a fellow of the World Academia of Science.

• 1. Basic principle of nonlocalities. 2. Basic of fractional operators. 3. Definitions of fractal-fractional operators with numerical approximations. 4. Error analysis. 5. Existence and uniqueness of fractal fractional differential equations. 6. A numerical solution of fractal-fractional ODE with linear interpolation. 7. Numerical scheme of fractal-fractional ODE with middle point interpolation. 8. Fractal-Fractional Euler method. 9. Application of fractal-fractional operators to a chaotic model. 10. Fractal-fractional Modified Chua chaotic attractor. 11. Application of fractal-fractional operators to study a new chaotic model. 12. Fractal-fractional operators and their application to a chaotic system with sinusoidal component. 13. Application of fractal-fractional operators to four-scroll chaotic system. 14. Application of fractal-fractional operators to a novel chaotic model. 15. A 4D chaotic system under fractal-fractional operators. 16. Self-excited and hidden attractors through fractal-fractional operators. 17. Dynamical analysis of a chaotic model in fractal-fractional operators. 18. A chaotic cancer model in fractal-fractional operators. 19. A multiple chaotic attractor model under fractal-fractional operators. 20. The dynamics of multiple chaotic attractor with fractal-fractional operators. 21. Dynamics of 3D chaotic systems with fractal-fractional operators. 22. The hidden attractors model with fractal-fractional operators. 23. An SIR epidemic model with Fractal-fractional derivative. 24. Application of fractal-fractional operators to COVID-19 infection.

Muhammad Altaf Khan completed his master’s degree in Applied mathematics from the Kohat University of Science in Technology in 2008. He completed his MS degree from Islamia college University Peshawar in 2012 in Applied mathematics and his research area is Mathematical Biology. He did his Ph.D. in Applied mathematics from Abdul Wali Khan University Mardan in 2015. He served as Lecturer in Abdul Wali Khan Mardan in the Department of Mathematics from 2015-2016 under the Umbrella of faculty development program. After that, he joined the City University of Science and Information Technology Peshawar and served there as Assistant Professor for the period August 2017-September 2019. He joined the post-doctorate position on October 01, 2019-December 2021 at the University of the Free State of South Africa. He also worked as a Researcher at Ton Duc Thang University Vietnam, March 2020-February 2021. He has been an Adjunct Professor at Airlangga University Indonesia for the period March 2021-Till February 2022. He is currently a Research Fellow at the University of the Free State of South Africa. He organized many special issues in reputable ISI journals, such as Advances in Difference equations, Results in Physics, Fractals, Alexandria Engineering Journals, Symmetry MDPI, etc. He is the author of more than 250 publications. He is an active editor in some reputable journals. He is an active researcher and worked working in fractional differential equations, evolutionary differential equations, optimal control problems, infectious diseases modeling, social sciences problems modeling, etc. He is a reviewer for many international journals of Springer, Wily, Hindawi, Elsevier, etc. Abdon Atangana works at the Institute for Groundwater Studies, University of the Free State, Bloemfontein, South Africa as a Full Professor. His research interests are, but are not limited to, fractional calculus and applications, numerical and analytical methods, and modeling. He is the author of more than 270 research papers and five books in top-tier journals of applied mathematics and groundwater modeling. He is the founder of numerous mathematical operators, for example, fractional differential and integral operators with singular and nonlocal kernels (Atangana-Baleanu derivatives and integral), fractal-fractional calculus, and piecewise calculus, and some numerical methods. He serves as an editor in top-tier journals in various fields of study. He has been invited as a plenary, and keynote speaker at more than 30 international conferences. He was elected Highly Cited Mathematician in 2019, Highly Cited Mathematician with Crossfield Impact in 2020, and Highly Cited Mathematician in 2021. He is a recipient of the Obada price 2020, the TWAS-Hamdan Award 2020, and many others. He is a fellow of the World Academia of Science.

• 1. Basic principle of nonlocalities. 2. Basic of fractional operators. 3. Definitions of fractal-fractional operators with numerical approximations. 4. Error analysis. 5. Existence and uniqueness of fractal fractional differential equations. 6. A numerical solution of fractal-fractional ODE with linear interpolation. 7. Numerical scheme of fractal-fractional ODE with middle point interpolation. 8. Fractal-Fractional Euler method. 9. Application of fractal-fractional operators to a chaotic model. 10. Fractal-fractional Modified Chua chaotic attractor. 11. Application of fractal-fractional operators to study a new chaotic model. 12. Fractal-fractional operators and their application to a chaotic system with sinusoidal component. 13. Application of fractal-fractional operators to four-scroll chaotic system. 14. Application of fractal-fractional operators to a novel chaotic model. 15. A 4D chaotic system under fractal-fractional operators. 16. Self-excited and hidden attractors through fractal-fractional operators. 17. Dynamical analysis of a chaotic model in fractal-fractional operators. 18. A chaotic cancer model in fractal-fractional operators. 19. A multiple chaotic attractor model under fractal-fractional operators. 20. The dynamics of multiple chaotic attractor with fractal-fractional operators. 21. Dynamics of 3D chaotic systems with fractal-fractional operators. 22. The hidden attractors model with fractal-fractional operators. 23. An SIR epidemic model with Fractal-fractional derivative. 24. Application of fractal-fractional operators to COVID-19 infection.