Article
Mathematics, Interdisciplinary Applications
Hussam Alrabaiah, Mati Ur Rahman, Ibrahim Mahariq, Samia Bushnaq, Muhammad Arfan
Summary: In this paper, a fractional mathematical model describing the co-infection of HBV and HCV is studied, and qualitative analysis and approximate solutions are obtained using fixed point theory. The proposed scheme is simulated at different fractional order values, showing that stability is achieved more rapidly at low orders.
FRACTALS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY
(2022)
Article
Materials Science, Multidisciplinary
Muhammad Arfan, Kamal Shah, Aman Ullah, Meshal Shutaywi, Poom Kumam, Zahir Shah
Summary: The research focuses on the mathematical model of the dynamics of tumor cells, immune cells, and drugs reaction systems, with qualitative analysis and approximate results obtained through numerical methods. The proposed model is applied to study the dynamics of tumor cells, aiding in the investigation of the dynamical interaction among tumor cells, immune cells, and drugs reactions in the disease.
RESULTS IN PHYSICS
(2021)
Article
Mathematics, Interdisciplinary Applications
Mohammadi Begum Jeelani, Abeer S. Alnahdi, Mohammed S. Abdo, Mohammed A. Almalahi, Nadiyah Hussain Alharthi, Kamal Shah
Summary: This work focuses on studying the transmission dynamics of CoV-2 in the presence of vaccination. Nonlinear analysis is used to determine the equilibrium points and R0 of the proposed model. The existence and uniqueness of solutions are investigated using fixed point theorems, while the stability of the model is analyzed using the Ulam-Hyers approach. Numerical analysis is performed using AB fractional calculus and Adams-Bashforth iterative numerical techniques. Real data from Saudi Arabia is used for graphical presentation of the respective results.
FRACTALS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY
(2023)
Article
Mathematics, Interdisciplinary Applications
Zareen A. Khan, Kamal Shah, Bahaaeldin Abdalla, Thabet Abdeljawad
Summary: This paper investigates the existence and stability of a chemostat model under fractal-fractional order derivative. The positivity and roundedness of the solution are examined, and the existence of a solution is found using the Banach and Schauder fixed-point theorems. A sufficient condition for the stability of solutions is obtained through numerical-functional analysis. The proposed system is proven to have a unique positive solution that satisfies the Ulam-Hyers and generalized U-H stability criteria. Numerical examples are provided to validate the theoretical results.
FRACTALS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY
(2023)
Article
Mathematics, Applied
Saeed M. Ali, Mohammed S. Abdo, Bhausaheb Sontakke, Kamal Shah, Thabet Abdeljawad
Summary: The paper introduces and analyzes a coupled system of second-order fractional pantograph differential equations with coupled four-point boundary conditions. Through the application of nonlinear alternatives and contraction mapping, the existence and uniqueness of solutions are proven. Additionally, the stability of the solutions is demonstrated using Ulam-Hyers stability theory.
Article
Mathematics, Applied
Ramzi B. Albadarneh, Iqbal Batiha, A. K. Alomari, Nedal Tahat
Summary: This work introduces a new power series formula to approximate the Caputo fractional-order operator, which is successfully used for approximate solutions of linear and nonlinear fractional-order differential equations. Numerical examples are provided to validate the formula.
Article
Mathematics, Interdisciplinary Applications
Zareen A. Khan, Kamal Shah, Bahaaeldin Abdalla, Thabet Abdeljawad
Summary: In this paper, the existence of numerical solution and stability of a chemostat model under fractal-fractional order derivative is studied. The positivity and roundedness of the solution are investigated, and the existence of a solution is found using the Banach and Schauder fixed-point theorems. A sufficient condition for the stability of solutions is obtained through numerical-functional analysis. The proposed system exists as a unique positive solution and satisfies the criteria of Ulam-Hyers and generalized U-H stability. Numerical examples are provided to verify the theoretical results. This research is important for understanding the hydrodynamics in porous media.
FRACTALS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY
(2023)
Article
Mathematics, Applied
Sachin Bhalekar, Prashant M. Gade
Summary: This study investigates the stability of synchronized fixed-point state for linear fractional-order coupled map lattice (CML). It is observed that the eigenvalues of the connectivity matrix determine the system's stability, similar to that in integer-order CML. Exact bounds are found for a one-dimensional lattice with translationally invariant coupling using the theory of circulant matrices, which can be extended to any finite dimension. Similar analysis can be conducted for the synchronized fixed point of nonlinear coupled fractional maps, where eigenvalues of the Jacobian matrix play the same role. The analysis demonstrates that the eigenvalues of the connectivity matrix play a pivotal role in the stability analysis of synchronized fixed point even in coupled fractional maps.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2022)
Article
Mathematics, Applied
Mustafa Ali Dokuyucu
Summary: This study expanded the analysis of the new finance chaotic model to Atangana-Baleanu-Caputo fractional derivative, investigated the existence solution using the fixed model theorem, and examined the uniqueness solution using the Sumudu transformation, providing a theoretical foundation for further research in the finance field.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
Mohammed S. Abdo, Thabet Abdeljawad, Kamal Shah, Saeed M. Ali
Summary: The paper develops and extends a qualitative analysis of a novel nonlinear system of fractional pantograph evolution differential equations, establishing essential conditions for existence and uniqueness results using fixed point techniques, and discussing Ulam-Hyers stability of solutions for the system. Two interesting pertinent examples are presented.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Mathematics
Xin Liu, Lili Chen, Yanfeng Zhao
Summary: This paper discusses the existence and uniqueness of solutions for a nonlinear fractional-order coupled delayed system with a new kind of boundary condition. The problem is transformed into an equivalent fixed point problem using the integral operator. A novel set of sufficient conditions that ensure the existence and uniqueness of solutions is derived by applying fixed point theorems. An example is provided to illustrate the effectiveness of the obtained results.
Article
Materials Science, Multidisciplinary
M. A. Almuqrin, P. Goswami, S. Sharma, I. Khan, R. S. Dubey, A. Khan
Summary: The paper extends the model of Ebola virus in bats to a fractional order mathematical model using the Atangana-Baleanu derivative operator. It presents a detailed proof for the existence, uniqueness, and stability of the solution for the fractional mathematical model, and utilizes a numerical approach to find the solution and represent the results graphically.
RESULTS IN PHYSICS
(2021)
Article
Mathematics
John R. Graef, Kadda Maazouz, Moussa Daif Allah Zaak
Summary: The authors investigate a nonlinear fractional pantograph boundary value problem with a variable order Hadamard fractional derivative and establish the existence and uniqueness results. This type of model is suitable for applications in strongly anomalous media. They also derive a generalized Lyapunov-type inequality for the problem and utilize fractional calculus and Krasnosel'skii's fixed point theorem in their proofs. An example is provided to illustrate their approach.
Article
Mathematics, Interdisciplinary Applications
Muhammad Sher, Aziz Khan, Kamal Shah, Thabet Abdeljawad
Summary: This paper investigates the analytical and approximate solutions of the sine-Gordon equation under the fractional-order derivatives. The Atangana-Baleanu-Caputo derivative and Modified Homotopy Perturbation Method are employed in this study, and the results are supported with graphical examples.
FRACTALS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY
(2023)
Article
Mathematics, Applied
Muhammed Jamil, Rahmat Ali Khan, Kamal Shah, Bahaaeldin Abdalla, Thaber Abdeljawad
Summary: This manuscript aims to investigate the existence theory of solution for a nonlinear boundary value problem of tripled system of fractional order hybrid sequential integro-differential equations. The analysis is based on results from fractional calculus and fixed point theory, leading to the generalization of Darbo's fixed point theorem and the establishment of Hyres-Ulam and generalized Hyres-Ulam stabilities results.
