Research

13 pages, 294 KiB

Open AccessArticle

Some Families of Differential Equations Associated with Multivariate Hermite Polynomials

by Badr Saad T. Alkahtani, Ibtehal Alazman and Shahid Ahmad Wani

Fractal Fract. 2023, 7(5), 390; https://doi.org/10.3390/fractalfract7050390 - 08 May 2023

Cited by 3 | Viewed by 856

In this article, the recurrence relations and shift operators for multivariate Hermite polynomials are derived using the factorization approach. Families of differential equations, including differential, integro–differential, and partial differential equations, are obtained using these operators. The Volterra integral for these polynomials is also [...] Read more.

In this article, the recurrence relations and shift operators for multivariate Hermite polynomials are derived using the factorization approach. Families of differential equations, including differential, integro–differential, and partial differential equations, are obtained using these operators. The Volterra integral for these polynomials is also discovered. Full article

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

18 pages, 1290 KiB

Open AccessArticle

Bifurcation and Stability of Two-Dimensional Activator–Inhibitor Model with Fractional-Order Derivative

by Messaoud Berkal and Mohammed Bakheet Almatrafi

Fractal Fract. 2023, 7(5), 344; https://doi.org/10.3390/fractalfract7050344 - 22 Apr 2023

Cited by 10 | Viewed by 1006

Abstract

In organisms’ bodies, the activities of enzymes can be catalyzed or inhibited by some inorganic and organic compounds. The interaction between enzymes and these compounds is successfully described by mathematics. The main purpose of this article is to investigate the dynamics of the [...] Read more.

In organisms’ bodies, the activities of enzymes can be catalyzed or inhibited by some inorganic and organic compounds. The interaction between enzymes and these compounds is successfully described by mathematics. The main purpose of this article is to investigate the dynamics of the activator–inhibitor system (Gierer–Meinhardt system), which is utilized to describe the interactions of chemical and biological phenomena. The system is considered with a fractional-order derivative, which is converted to an ordinary derivative using the definition of the conformable fractional derivative. The obtained differential equations are solved using the separation of variables. The stability of the obtained positive equilibrium point of this system is analyzed and discussed. We find that this point can be locally asymptotically stable, a source, a saddle, or non-hyperbolic under certain conditions. Moreover, this article concentrates on exploring a Neimark–Sacker bifurcation and a period-doubling bifurcation. Then, we present some numerical computations to verify the obtained theoretical results. The findings of this work show that the governing system undergoes the Neimark–Sacker bifurcation and the period-doubling bifurcation under certain conditions. These types of bifurcation occur in small domains, as shown theoretically and numerically. Some 2D figures are illustrated to visualize the behavior of the solutions in some domains. Full article

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

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19 pages, 930 KiB

Open AccessArticle

Dynamical Analysis of Generalized Tumor Model with Caputo Fractional-Order Derivative

by Ausif Padder, Laila Almutairi, Sania Qureshi, Amanullah Soomro, Afroz Afroz, Evren Hincal and Asifa Tassaddiq

Fractal Fract. 2023, 7(3), 258; https://doi.org/10.3390/fractalfract7030258 - 13 Mar 2023

Cited by 21 | Viewed by 1920

Abstract

In this study, we perform a dynamical analysis of a generalized tumor model using the Caputo fractional-order derivative. Tumor growth models are widely used in biomedical research to understand the dynamics of tumor development and to evaluate potential treatments. The Caputo fractional-order derivative [...] Read more.

In this study, we perform a dynamical analysis of a generalized tumor model using the Caputo fractional-order derivative. Tumor growth models are widely used in biomedical research to understand the dynamics of tumor development and to evaluate potential treatments. The Caputo fractional-order derivative is a mathematical tool that is recently being applied to model biological systems, including tumor growth. We present a detailed mathematical analysis of the generalized tumor model with the Caputo fractional-order derivative and examine its dynamical behavior. Our results show that the Caputo fractional-order derivative provides a more accurate description of the tumor growth dynamics compared to classical integer-order derivatives. We also provide a comprehensive stability analysis of the tumor model and show that the fractional-order derivative allows for a more nuanced understanding of the stability of the system. The least-square curve fitting method fits several biological parameters, including the fractional-order parameter

α

. In conclusion, our study provides new insights into the dynamics of tumor growth and highlights the potential of the Caputo fractional-order derivative as a valuable tool in biomedical research. The results of this study shall have significant implications for the development of more effective treatments for tumor growth and the design of more accurate mathematical models of tumor development. Full article

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

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10 pages, 300 KiB

Open AccessArticle

Certain Properties and Applications of Δ_h Hybrid Special Polynomials Associated with Appell Sequences

by Rabab Alyusof and Shahid Ahmmad Wani

Fractal Fract. 2023, 7(3), 233; https://doi.org/10.3390/fractalfract7030233 - 06 Mar 2023

Cited by 3 | Viewed by 862

Abstract

The development of certain aspects of special polynomials in line with the monomiality principle, operational rules, and other properties and their aspects is obvious and indisputable. The study presented in this paper follows this line of research. By using the monomiality principle, new [...] Read more.

The development of certain aspects of special polynomials in line with the monomiality principle, operational rules, and other properties and their aspects is obvious and indisputable. The study presented in this paper follows this line of research. By using the monomiality principle, new outcomes are produced, and their differential equation and series representation is obtained, which are important in several branches of mathematics and physics. Thus, in line with prior facts, our aim is to introduce the

Δ_{h}

hybrid special polynomials associated with Hermite polynomials denoted by

_{Δ_{h} H} Q_{m} (u, v, w; h)

. Further, we obtain some well-known main properties and explicit forms satisfied by these polynomials. Full article

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

32 pages, 929 KiB

Open AccessArticle

Existence Results for Caputo Tripled Fractional Differential Inclusions with Integral and Multi-Point Boundary Conditions

by Muath Awadalla and Murugesan Manigandan

Fractal Fract. 2023, 7(2), 182; https://doi.org/10.3390/fractalfract7020182 - 12 Feb 2023

Cited by 1 | Viewed by 837

Abstract

In this study, based on Coitz and Nadler’s fixed point theorem and the non-linear alternative for Kakutani maps, existence results for a tripled system of sequential fractional differential inclusions (SFDIs) with integral and multi-point boundary conditions (BCs) in investigated. A practical examples are [...] Read more.

In this study, based on Coitz and Nadler’s fixed point theorem and the non-linear alternative for Kakutani maps, existence results for a tripled system of sequential fractional differential inclusions (SFDIs) with integral and multi-point boundary conditions (BCs) in investigated. A practical examples are given to illustrate the obtained the theoretical results. Full article

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

15 pages, 3201 KiB

Open AccessArticle

Bifurcation and Analytical Solutions of the Space-Fractional Stochastic Schrödinger Equation with White Noise

by Muneerah Al Nuwairan

Fractal Fract. 2023, 7(2), 157; https://doi.org/10.3390/fractalfract7020157 - 05 Feb 2023

Cited by 7 | Viewed by 1123

Abstract

The qualitative theory for planar dynamical systems is used to study the bifurcation of the wave solutions for the space-fractional nonlinear Schrödinger equation with multiplicative white noise. Employing the first integral, we introduce some new wave solutions, assorted into periodic, solitary, and kink [...] Read more.

The qualitative theory for planar dynamical systems is used to study the bifurcation of the wave solutions for the space-fractional nonlinear Schrödinger equation with multiplicative white noise. Employing the first integral, we introduce some new wave solutions, assorted into periodic, solitary, and kink wave solutions. The dependence of the solutions on the initial conditions is investigated. Some solutions are clarified by the display of their 2D and 3D representations with varying levels of noise to show the influence of multiplicative white noise on the solutions. Full article

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

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12 pages, 3198 KiB

Open AccessArticle

New Analytical Solutions for Time-Fractional Stochastic (3+1)-Dimensional Equations for Fluids with Gas Bubbles and Hydrodynamics

by Mohammed Alhamud, Mamdouh Elbrolosy and Adel Elmandouh

Fractal Fract. 2023, 7(1), 16; https://doi.org/10.3390/fractalfract7010016 - 25 Dec 2022

Cited by 7 | Viewed by 967

Abstract

This paper explores the effects of spatial fractional derivatives and the multiplicative Wiener process on the analytical solutions for (3+1)-dimensional fractional stochastic equations for fluids with gas bubbles. We study the bifurcation of the analytical solutions and introduce new fractional stochastic solutions. We [...] Read more.

This paper explores the effects of spatial fractional derivatives and the multiplicative Wiener process on the analytical solutions for (3+1)-dimensional fractional stochastic equations for fluids with gas bubbles. We study the bifurcation of the analytical solutions and introduce new fractional stochastic solutions. We also discuss how the solutions differ depending on the initial conditions. The new solutions are notably more beneficial and impactful for understanding various, significant, and incredibly hard physical phenomena due to the significance of the modified fractional stochastic (3+1)-dimensional equations for fluids with gas bubbles and hydrodynamics. We also discuss the effects of the fractional order and the Wiener process on the obtained analytical solutions. Full article

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

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15 pages, 3803 KiB

Open AccessArticle

Bifurcation of Exact Solutions for the Space-Fractional Stochastic Modified Benjamin–Bona–Mahony Equation

by Adel Elmandouh and Emad Fadhal

Fractal Fract. 2022, 6(12), 718; https://doi.org/10.3390/fractalfract6120718 - 02 Dec 2022

Cited by 9 | Viewed by 1104

Abstract

This paper studies the influence of space-fractional and multiplicative noise on the exact solutions of the space-fractional stochastic dispersive modified Benjamin–Bona–Mahony equation, driven in Ito’s sense by a multiplicative Wiener process. The bifurcation of the exact solutions is investigated, and novel fractional stochastic [...] Read more.

This paper studies the influence of space-fractional and multiplicative noise on the exact solutions of the space-fractional stochastic dispersive modified Benjamin–Bona–Mahony equation, driven in Ito’s sense by a multiplicative Wiener process. The bifurcation of the exact solutions is investigated, and novel fractional stochastic solutions are presented. The dependence of the solutions on the initial conditions is discussed. Due to the significance of the fractional stochastic modified Benjamin–Bona–Mahony equation in describing the propagation of surface long waves in nonlinear dispersive media, the derived solutions are significantly more helpful for and influential in comprehending diverse, crucial, and challenging physical phenomena. The effect of the Wiener process and the fractional order on the exact solutions are studied. Full article

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

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33 pages, 4987 KiB

Open AccessArticle

A Four Step Feedback Iteration and Its Applications in Fractals

by Asifa Tassaddiq, Muhammad Tanveer, Muhammad Azhar, Waqas Nazeer and Sania Qureshi

Fractal Fract. 2022, 6(11), 662; https://doi.org/10.3390/fractalfract6110662 - 09 Nov 2022

Cited by 5 | Viewed by 1901

Abstract

Fractals play a vital role in modeling the natural environment. The present aim is to investigate the escape criterion to generate specific fractals such as Julia sets, Mandelbrot sets and Multi-corns via F-iteration using complex functions [...] Read more.

Fractals play a vital role in modeling the natural environment. The present aim is to investigate the escape criterion to generate specific fractals such as Julia sets, Mandelbrot sets and Multi-corns via F-iteration using complex functions

h (z) = z^{n} + c

,

h (z) = sin (z^{n}) + c

and

h (z) = e^{z^{n}} + c

,

n \geq 2, c \in C

. We observed some beautiful Julia sets, Mandelbrot sets and Multi-corns for n = 2, 3 and 4. We generalize the algorithms of the Julia set and Mandelbrot set to visualize some Julia sets, Mandelbrot sets and Multi-corns. Moreover, we calculate image generation time in seconds at different values of input parameters. Full article

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

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