Fractal and Fractional

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4 pages, 203 KiB

Open AccessEditorial

Editorial for Special Issue “New Advancements in Pure and Applied Mathematics via Fractals and Fractional Calculus”

by Asifa Tassaddiq and Muhammad Yaseen

Fractal Fract. 2022, 6(6), 284; https://doi.org/10.3390/fractalfract6060284 - 25 May 2022

Viewed by 1499

Abstract

Fractional calculus has reshaped science and technology since its first appearance in a letter received to Gottfried Wilhelm Leibniz from Guil-laume de l’Hôpital in the year 1695 [...] Full article

(This article belongs to the Special Issue New Advancements in Pure and Applied Mathematics via Fractals and Fractional Calculus)

Research

Jump to: Editorial

19 pages, 538 KiB

Open AccessArticle

Certain Hybrid Matrix Polynomials Related to the Laguerre-Sheffer Family

by Tabinda Nahid and Junesang Choi

Fractal Fract. 2022, 6(4), 211; https://doi.org/10.3390/fractalfract6040211 - 9 Apr 2022

Cited by 9 | Viewed by 1602

Abstract

The main goal of this article is to explore a new type of polynomials, specifically the Gould-Hopper-Laguerre-Sheffer matrix polynomials, through operational techniques. The generating function and operational representations for this new family of polynomials will be established. In addition, these specific matrix polynomials [...] Read more.

The main goal of this article is to explore a new type of polynomials, specifically the Gould-Hopper-Laguerre-Sheffer matrix polynomials, through operational techniques. The generating function and operational representations for this new family of polynomials will be established. In addition, these specific matrix polynomials are interpreted in terms of quasi-monomiality. The extended versions of the Gould-Hopper-Laguerre-Sheffer matrix polynomials are introduced, and their characteristics are explored using the integral transform. Further, examples of how these results apply to specific members of the matrix polynomial family are shown. Full article

(This article belongs to the Special Issue New Advancements in Pure and Applied Mathematics via Fractals and Fractional Calculus)

19 pages, 368 KiB

Open AccessArticle

Some q-Fractional Estimates of Trapezoid like Inequalities Involving Raina’s Function

by Kamsing Nonlaopon, Muhammad Uzair Awan, Muhammad Zakria Javed, Hüseyin Budak and Muhammad Aslam Noor

Fractal Fract. 2022, 6(4), 185; https://doi.org/10.3390/fractalfract6040185 - 25 Mar 2022

Cited by 3 | Viewed by 1660

Abstract

In this paper, we derive two new identities involving q-Riemann-Liouville fractional integrals. Using these identities, as auxiliary results, we derive some new q-fractional estimates of trapezoidal-like inequalities, essentially using the class of generalized exponential convex functions. Full article

(This article belongs to the Special Issue New Advancements in Pure and Applied Mathematics via Fractals and Fractional Calculus)

13 pages, 325 KiB

Open AccessArticle

Extremal Solutions of Generalized Caputo-Type Fractional-Order Boundary Value Problems Using Monotone Iterative Method

by Choukri Derbazi, Zidane Baitiche, Mohammed S. Abdo, Kamal Shah, Bahaaeldin Abdalla and Thabet Abdeljawad

Fractal Fract. 2022, 6(3), 146; https://doi.org/10.3390/fractalfract6030146 - 7 Mar 2022

Cited by 18 | Viewed by 2391

Abstract

The aim of this research work is to derive some appropriate results for extremal solutions to a class of generalized Caputo-type nonlinear fractional differential equations (FDEs) under nonlinear boundary conditions (NBCs). The aforesaid results are derived by using the monotone iterative method, which [...] Read more.

The aim of this research work is to derive some appropriate results for extremal solutions to a class of generalized Caputo-type nonlinear fractional differential equations (FDEs) under nonlinear boundary conditions (NBCs). The aforesaid results are derived by using the monotone iterative method, which exercises the procedure of upper and lower solutions. Two sequences of extremal solutions are generated in which one converges to the upper and the other to the corresponding lower solution. The method does not need any prior discretization or collocation for generating the aforesaid two sequences for upper and lower solutions. Further, the aforesaid techniques produce a fruitful combination of upper and lower solutions. To demonstrate our results, we provide some pertinent examples. Full article

(This article belongs to the Special Issue New Advancements in Pure and Applied Mathematics via Fractals and Fractional Calculus)

17 pages, 1025 KiB

Open AccessArticle

Numerical Analysis of Time-Fractional Whitham-Broer-Kaup Equations with Exponential-Decay Kernel

by Humaira Yasmin

Fractal Fract. 2022, 6(3), 142; https://doi.org/10.3390/fractalfract6030142 - 2 Mar 2022

Cited by 14 | Viewed by 2471

Abstract

This paper presents the semi-analytical analysis of the fractional-order non-linear coupled system of Whitham-Broer-Kaup equations. An iterative process is designed to analyze analytical findings to the specified non-linear partial fractional derivatives scheme utilizing the Yang transformation coupled with the Adomian technique. The fractional [...] Read more.

This paper presents the semi-analytical analysis of the fractional-order non-linear coupled system of Whitham-Broer-Kaup equations. An iterative process is designed to analyze analytical findings to the specified non-linear partial fractional derivatives scheme utilizing the Yang transformation coupled with the Adomian technique. The fractional derivative is considered in the sense of Caputo-Fabrizio. Two numerical problems show the suggested method. Moreover, the results of the suggested technique are compared with the solution of other well-known numerical techniques such as the Homotopy perturbation technique, Adomian decomposition technique, and the Variation iteration technique. Numerical simulation has been carried out to verify that the suggested methodologies are accurate and reliable, and the results are revealed using graphs and tables. Comparing the analytical and actual solutions demonstrates that the proposed approaches effectively solve complicated non-linear problems. Furthermore, the proposed methodologies control and manipulate the achieved numerical solutions in a vast acceptable region in an extreme manner. It will provide us with a simple process to control and adjust the convergence region of the series solution. Full article

(This article belongs to the Special Issue New Advancements in Pure and Applied Mathematics via Fractals and Fractional Calculus)

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20 pages, 431 KiB

Open AccessArticle

Numerical Study of Caputo Fractional-Order Differential Equations by Developing New Operational Matrices of Vieta–Lucas Polynomials

by Zulfiqar Ahmad Noor, Imran Talib, Thabet Abdeljawad and Manar A. Alqudah

Fractal Fract. 2022, 6(2), 79; https://doi.org/10.3390/fractalfract6020079 - 31 Jan 2022

Cited by 8 | Viewed by 2620

Abstract

In this article, we develop a numerical method based on the operational matrices of shifted Vieta–Lucas polynomials (VLPs) for solving Caputo fractional-order differential equations (FDEs). We derive a new operational matrix of the fractional-order derivatives in the Caputo sense, which is then used [...] Read more.

In this article, we develop a numerical method based on the operational matrices of shifted Vieta–Lucas polynomials (VLPs) for solving Caputo fractional-order differential equations (FDEs). We derive a new operational matrix of the fractional-order derivatives in the Caputo sense, which is then used with spectral tau and spectral collocation methods to reduce the FDEs to a system of algebraic equations. Several numerical examples are given to show the accuracy of this method. These examples show that the obtained results have good agreement with the analytical solutions in both linear and non-linear FDEs. In addition to this, the numerical results obtained by using our method are compared with the numerical results obtained otherwise in the literature. Full article

(This article belongs to the Special Issue New Advancements in Pure and Applied Mathematics via Fractals and Fractional Calculus)

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17 pages, 314 KiB

Open AccessArticle

Hermite-Hadamard Fractional Inequalities for Differentiable Functions

by Muhammad Samraiz, Zahida Perveen, Gauhar Rahman, Muhammad Adil Khan and Kottakkaran Sooppy Nisar

Fractal Fract. 2022, 6(2), 60; https://doi.org/10.3390/fractalfract6020060 - 25 Jan 2022

Cited by 10 | Viewed by 2053

Abstract

In this article, we look at a variety of mean-type integral inequalities for a well-known Hilfer fractional derivative. We consider twice differentiable convex and s-convex functions for

s \in (0, 1]

that have applications in optimization theory. In order [...] Read more.

In this article, we look at a variety of mean-type integral inequalities for a well-known Hilfer fractional derivative. We consider twice differentiable convex and s-convex functions for

s \in (0, 1]

that have applications in optimization theory. In order to infer more interesting mean inequalities, some identities are also established. The consequences for Caputo fractional derivative are presented as special cases to our general conclusions. Full article

(This article belongs to the Special Issue New Advancements in Pure and Applied Mathematics via Fractals and Fractional Calculus)

22 pages, 526 KiB

Open AccessArticle

Comparative Numerical Study of Spline-Based Numerical Techniques for Time Fractional Cattaneo Equation in the Sense of Caputo–Fabrizio

by Muhammad Yaseen, Qamar Un Nisa Arif, Reny George and Sana Khan

Fractal Fract. 2022, 6(2), 50; https://doi.org/10.3390/fractalfract6020050 - 18 Jan 2022

Cited by 4 | Viewed by 2159

Abstract

This study focuses on numerically addressing the time fractional Cattaneo equation involving Caputo–Fabrizio derivative using spline-based numerical techniques. The splines used are the cubic B-splines, trigonometric cubic B-splines and extended cubic B-splines. The space derivative is approximated using B-splines basis functions, Caputo–Fabrizio derivative [...] Read more.

This study focuses on numerically addressing the time fractional Cattaneo equation involving Caputo–Fabrizio derivative using spline-based numerical techniques. The splines used are the cubic B-splines, trigonometric cubic B-splines and extended cubic B-splines. The space derivative is approximated using B-splines basis functions, Caputo–Fabrizio derivative is discretized, using a finite difference approach. The techniques are also put through a stability analysis to verify that the errors do not pile up. The proposed scheme’s convergence analysis is also explored. The key advantage of the schemes is that the approximation solution is produced as a smooth piecewise continuous function, allowing us to approximate a solution at any place in the domain of interest. A numerical study is performed using various splines, and the outcomes are compared to demonstrate the efficiency of the proposed schemes. Full article

(This article belongs to the Special Issue New Advancements in Pure and Applied Mathematics via Fractals and Fractional Calculus)

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21 pages, 4996 KiB

Open AccessArticle

Discretization, Bifurcation, and Control for a Class of Predator-Prey Interactions

by Asifa Tassaddiq, Muhammad Sajjad Shabbir, Qamar Din and Humera Naaz

Fractal Fract. 2022, 6(1), 31; https://doi.org/10.3390/fractalfract6010031 - 6 Jan 2022

Cited by 27 | Viewed by 2329

Abstract

The present study focuses on the dynamical aspects of a discrete-time Leslie-Gower predator-prey model accompanied by a Holling type III functional response. Discretization is conducted by applying a piecewise constant argument method of differential equations. Moreover, boundedness, existence, uniqueness, and a local stability [...] Read more.

The present study focuses on the dynamical aspects of a discrete-time Leslie-Gower predator-prey model accompanied by a Holling type III functional response. Discretization is conducted by applying a piecewise constant argument method of differential equations. Moreover, boundedness, existence, uniqueness, and a local stability analysis of biologically feasible equilibria were investigated. By implementing the center manifold theorem and bifurcation theory, our study reveals that the given system undergoes period-doubling and Neimark-Sacker bifurcation around the interior equilibrium point. By contrast, chaotic attractors ensure chaos. To avoid these unpredictable situations, we establish a feedback-control strategy to control the chaos created under the influence of bifurcation. The fractal dimensions of the proposed model are calculated. The maximum Lyapunov exponents and phase portraits are depicted to further confirm the complexity and chaotic behavior. Finally, numerical simulations are presented to confirm the theoretical and analytical findings. Full article

(This article belongs to the Special Issue New Advancements in Pure and Applied Mathematics via Fractals and Fractional Calculus)

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25 pages, 1725 KiB

Open AccessArticle

A New Three-Step Root-Finding Numerical Method and Its Fractal Global Behavior

by Asifa Tassaddiq, Sania Qureshi, Amanullah Soomro, Evren Hincal, Dumitru Baleanu and Asif Ali Shaikh

Fractal Fract. 2021, 5(4), 204; https://doi.org/10.3390/fractalfract5040204 - 8 Nov 2021

Cited by 15 | Viewed by 2577

Abstract

There is an increasing demand for numerical methods to obtain accurate approximate solutions for nonlinear models based upon polynomials and transcendental equations under both single and multivariate variables. Keeping in mind the high demand within the scientific literature, we attempt to devise a [...] Read more.

There is an increasing demand for numerical methods to obtain accurate approximate solutions for nonlinear models based upon polynomials and transcendental equations under both single and multivariate variables. Keeping in mind the high demand within the scientific literature, we attempt to devise a new nonlinear three-step method with tenth-order convergence while using six functional evaluations (three functions and three first-order derivatives) per iteration. The method has an efficiency index of about 1.4678, which is higher than most optimal methods. Convergence analysis for single and systems of nonlinear equations is also carried out. The same is verified with the approximated computational order of convergence in the absence of an exact solution. To observe the global fractal behavior of the proposed method, different types of complex functions are considered under basins of attraction. When compared with various well-known methods, it is observed that the proposed method achieves prespecified tolerance in the minimum number of iterations while assuming different initial guesses. Nonlinear models include those employed in science and engineering, including chemical, electrical, biochemical, geometrical, and meteorological models. Full article

(This article belongs to the Special Issue New Advancements in Pure and Applied Mathematics via Fractals and Fractional Calculus)

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31 pages, 2202 KiB

Open AccessArticle

Fractional Dynamics of Typhoid Fever Transmission Models with Mass Vaccination Perspectives

by Hamadjam Abboubakar, Raissa Kom Regonne and Kottakkaran Sooppy Nisar

Fractal Fract. 2021, 5(4), 149; https://doi.org/10.3390/fractalfract5040149 - 30 Sep 2021

Cited by 11 | Viewed by 2093

Abstract

In this work, we formulate and mathematically study integer and fractional models of typhoid fever transmission dynamics. The models include vaccination as a control measure. After recalling some preliminary results for the integer model (determination of the epidemiological threshold denoted by

R_{c}

[...] Read more.

In this work, we formulate and mathematically study integer and fractional models of typhoid fever transmission dynamics. The models include vaccination as a control measure. After recalling some preliminary results for the integer model (determination of the epidemiological threshold denoted by

R_{c}

, asymptotic stability of the equilibrium point without disease whenever

R_{c} < 1

, the existence of an equilibrium point with disease whenever

R_{c} > 1

), we replace the integer derivative with the Caputo derivative. We perform a stability analysis of the disease-free equilibrium and prove the existence and uniqueness of the solution of the fractional model using fixed point theory. We construct the numerical scheme and prove its stability. Simulation results show that when the fractional-order

η

decreases, the peak of infected humans is delayed. To reduce the proliferation of the disease, mass vaccination combined with environmental sanitation is recommended. We then extend the previous model by replacing the mass action incidences with standard incidences. We compute the corresponding epidemiological threshold denoted by

R_{c ⋆}

and ensure the uniform stability of the disease-free equilibrium, for both new models, when

R_{c ⋆} < 1

. A new calibration of the new model is conducted with real data of Mbandjock, Cameroon, to estimate

R_{c ⋆} = 1.4348

. We finally perform several numerical simulations that permit us to conclude that such diseases can possibly be tackled through vaccination combined with environmental sanitation. Full article

(This article belongs to the Special Issue New Advancements in Pure and Applied Mathematics via Fractals and Fractional Calculus)

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18 pages, 338 KiB

Open AccessArticle

On Weighted (k, s)-Riemann-Liouville Fractional Operators and Solution of Fractional Kinetic Equation

by Muhammad Samraiz, Muhammad Umer, Artion Kashuri, Thabet Abdeljawad, Sajid Iqbal and Nabil Mlaiki

Fractal Fract. 2021, 5(3), 118; https://doi.org/10.3390/fractalfract5030118 - 13 Sep 2021

Cited by 15 | Viewed by 2034

Abstract

In this article, we establish the weighted

(k, s)

-Riemann-Liouville fractional integral and differential operators. Some certain properties of the operators and the weighted generalized Laplace transform of the new operators are part of the paper. The article consists of [...] Read more.

In this article, we establish the weighted

(k, s)

-Riemann-Liouville fractional integral and differential operators. Some certain properties of the operators and the weighted generalized Laplace transform of the new operators are part of the paper. The article consists of Chebyshev-type inequalities involving a weighted fractional integral. We propose an integro-differential kinetic equation using the novel fractional operators and find its solution by applying weighted generalized Laplace transforms. Full article

(This article belongs to the Special Issue New Advancements in Pure and Applied Mathematics via Fractals and Fractional Calculus)

18 pages, 355 KiB

Open AccessEditor’s ChoiceArticle

On the General Solutions of Some Non-Homogeneous Div-Curl Systems with Riemann–Liouville and Caputo Fractional Derivatives

by Briceyda B. Delgado and Jorge E. Macías-Díaz

Fractal Fract. 2021, 5(3), 117; https://doi.org/10.3390/fractalfract5030117 - 10 Sep 2021

Cited by 20 | Viewed by 1908

Abstract

In this work, we investigate analytically the solutions of a nonlinear div-curl system with fractional derivatives of the Riemann–Liouville or Caputo types. To this end, the fractional-order vector operators of divergence, curl and gradient are identified as components of the fractional Dirac operator [...] Read more.

In this work, we investigate analytically the solutions of a nonlinear div-curl system with fractional derivatives of the Riemann–Liouville or Caputo types. To this end, the fractional-order vector operators of divergence, curl and gradient are identified as components of the fractional Dirac operator in quaternionic form. As one of the most important results of this manuscript, we derive general solutions of some non-homogeneous div-curl systems that consider the presence of fractional-order derivatives of the Riemann–Liouville or Caputo types. A fractional analogous to the Teodorescu transform is presented in this work, and we employ some properties of its component operators, developed in this work to establish a generalization of the Helmholtz decomposition theorem in fractional space. Additionally, right inverses of the fractional-order curl, divergence and gradient vector operators are obtained using Riemann–Liouville and Caputo fractional operators. Finally, some consequences of these results are provided as applications at the end of this work. Full article

(This article belongs to the Special Issue New Advancements in Pure and Applied Mathematics via Fractals and Fractional Calculus)

31 pages, 1811 KiB

Open AccessArticle

Analytic Fuzzy Formulation of a Time-Fractional Fornberg–Whitham Model with Power and Mittag–Leffler Kernels

by Saima Rashid, Rehana Ashraf, Ahmet Ocak Akdemir, Manar A. Alqudah, Thabet Abdeljawad and Mohamed S. Mohamed

Fractal Fract. 2021, 5(3), 113; https://doi.org/10.3390/fractalfract5030113 - 8 Sep 2021

Cited by 12 | Viewed by 2196

Abstract

This manuscript assesses a semi-analytical method in connection with a new hybrid fuzzy integral transform and the Adomian decomposition method via the notion of fuzziness known as the Elzaki Adomian decomposition method (briefly, EADM). Moreover, we use the aforesaid strategy to address the [...] Read more.

This manuscript assesses a semi-analytical method in connection with a new hybrid fuzzy integral transform and the Adomian decomposition method via the notion of fuzziness known as the Elzaki Adomian decomposition method (briefly, EADM). Moreover, we use the aforesaid strategy to address the time-fractional Fornberg–Whitham equation (FWE) under

g H

-differentiability by employing different initial conditions (IC). Several algebraic aspects of the fuzzy Caputo fractional derivative (CFD) and fuzzy Atangana–Baleanu (AB) fractional derivative operator in the Caputo sense, with respect to the Elzaki transform, are presented to validate their utilities. Apart from that, a general algorithm for fuzzy Caputo and AB fractional derivatives in the Caputo sense is proposed. Some illustrative cases are demonstrated to understand the algorithmic approach of FWE. Taking into consideration the uncertainty parameter

ζ \in [0, 1]

and various fractional orders, the convergence and error analysis are reported by graphical representations of FWE that have close harmony with the closed form solutions. It is worth mentioning that the projected approach to fuzziness is to verify the supremacy and reliability of configuring numerical solutions to nonlinear fuzzy fractional partial differential equations arising in physical and complex structures. Full article

(This article belongs to the Special Issue New Advancements in Pure and Applied Mathematics via Fractals and Fractional Calculus)

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29 pages, 3394 KiB

Open AccessArticle

Novel Computations of the Time-Fractional Fisher’s Model via Generalized Fractional Integral Operators by Means of the Elzaki Transform

by Saima Rashid, Zakia Hammouch, Hassen Aydi, Abdulaziz Garba Ahmad and Abdullah M. Alsharif

Fractal Fract. 2021, 5(3), 94; https://doi.org/10.3390/fractalfract5030094 - 12 Aug 2021

Cited by 40 | Viewed by 3194

Abstract

The present investigation dealing with a hybrid technique coupled with a new iterative transform method, namely the iterative Elzaki transform method (IETM), is employed to solve the nonlinear fractional Fisher’s model. Fisher’s equation is a precise mathematical result that arose in population dynamics [...] Read more.

The present investigation dealing with a hybrid technique coupled with a new iterative transform method, namely the iterative Elzaki transform method (IETM), is employed to solve the nonlinear fractional Fisher’s model. Fisher’s equation is a precise mathematical result that arose in population dynamics and genetics, specifically in chemistry. The Caputo and Antagana-Baleanu fractional derivatives in the Caputo sense are used to test the intricacies of this mechanism numerically. In order to examine the approximate findings of fractional-order Fisher’s type equations, the IETM solutions are obtained in series representation. Moreover, the stability of the approach was demonstrated using fixed point theory. Several illustrative cases are described that strongly agree with the precise solutions. Moreover, tables and graphs are included in order to conceptualize the influence of the fractional order and on the previous findings. The projected technique illustrates that only a few terms are sufficient for finding an approximate outcome, which is computationally appealing and accurate to analyze. Additionally, the offered procedure is highly robust, explicit, and viable for nonlinear fractional PDEs, but it could be generalized to other complex physical phenomena. Full article

(This article belongs to the Special Issue New Advancements in Pure and Applied Mathematics via Fractals and Fractional Calculus)

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12 pages, 306 KiB

Open AccessArticle

Some New Harmonically Convex Function Type Generalized Fractional Integral Inequalities

by Rana Safdar Ali, Aiman Mukheimer, Thabet Abdeljawad, Shahid Mubeen, Sabila Ali, Gauhar Rahman and Kottakkaran Sooppy Nisar

Fractal Fract. 2021, 5(2), 54; https://doi.org/10.3390/fractalfract5020054 - 7 Jun 2021

Cited by 11 | Viewed by 2577

Abstract

In this article, we established a new version of generalized fractional Hadamard and Fejér–Hadamard type integral inequalities. A fractional integral operator (FIO) with a non-singular function (multi-index Bessel function) as its kernel and monotone increasing functions is utilized to obtain the new version [...] Read more.

In this article, we established a new version of generalized fractional Hadamard and Fejér–Hadamard type integral inequalities. A fractional integral operator (FIO) with a non-singular function (multi-index Bessel function) as its kernel and monotone increasing functions is utilized to obtain the new version of such fractional inequalities. Our derived results are a generalized form of several proven inequalities already existing in the literature. The proven inequalities are useful for studying the stability and control of corresponding fractional dynamic equations. Full article

(This article belongs to the Special Issue New Advancements in Pure and Applied Mathematics via Fractals and Fractional Calculus)

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