Matematik Bölümü Yayın Koleksiyonu

Permanent URI for this collectionhttps://hdl.handle.net/20.500.12416/413

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Citation - Scopus: 23
A Caputo-Fabrizio Fractional-Order Cholera Model And İts Sensitivity Analysis
(Mehmet Yavuz, 2023) Ahmed, I.; Jarad, Fahd; Akgül, A.; Jarad, F.; Kumam, P.; Nonlaopon, K.; 234808; Matematik
In recent years, the availability of advanced computational techniques has led to a growing emphasis on fractional-order derivatives. This development has enabled researchers to explore the intricate dynamics of various biological models by employing fractional-order derivatives instead of traditional integer-order derivatives. This paper proposes a Caputo-Fabrizio fractional-order cholera epidemic model. Fixed-point theorems are utilized to investigate the existence and uniqueness of solutions. A recent and effective numerical scheme is employed to demonstrate the model’s complex behaviors and highlight the advantages of fractional-order derivatives. Additionally, a sensitivity analysis is conducted to identify the most influential parameters. © 2023 by the authors.
Citation - WoS: 9
Citation - Scopus: 9
A class of fractal Hilbert-type inequalities obtained via Cantor-type spherical coordinates
(Wiley, 2021) Baleanu, Dumitru; Baleanu, Dumitru; Krnic, Mario; Vukovic, Predrag; 56389; Matematik
We present a class of higher dimensional Hilbert-type inequalities on a fractal set (Double-struck capital R+alpha n)k. The crucial step in establishing our results are higher dimensional spherical coordinates on a fractal space. Further, we impose the corresponding conditions under which the constants appearing in the established Hilbert-type inequalities are the best possible. As an application, our results are compared with the previous results known from the literature.
Citation - WoS: 4
Citation - Scopus: 4
A Class of Refinement Schemes With Two Shape Control Parameters
(Ieee-inst Electrical Electronics Engineers inc, 2020) Mustafa, Ghulam; Baleanu, Dumitru; Hameed, Rabia; Baleanu, Dumitru; Mahmood, Ayesha; 56389; Matematik
A subdivision scheme defines a smooth curve or surface as the limit of a sequence of successive refinements of given polygon or mesh. These schemes take polygons or meshes as inputs and produce smooth curves or surfaces as outputs. In this paper, a class of combine refinement schemes with two shape control parameters is presented. These even and odd rules of these schemes have complexity three and four respectively. The even rule is designed to modify the vertices of the given polygon, whereas the odd rule is designed to insert a new point between every edge of the given polygon. These schemes can produce high order of continuous shapes than existing combine binary and ternary family of schemes. It has been observed that the schemes have interpolating and approximating behaviors depending on the values of parameters. These schemes have an interproximate behavior in the case of non-uniform setting of the parameters. These schemes can be considered as the generalized version of some of the interpolating and B-spline schemes. The theoretical as well as the numerical and graphical analysis of the shapes produced by these schemes are also presented.
Citation - WoS: 16
A comparative study of silicon nitride and SiAlON ceramics against E. coli
(Elsevier Sci Ltd, 2021) Akin, Seniz R. Kushan; Garcia, Caterina Bartomeu; Webster, Thomas J.; 224219
In recent decades, due to some limitations from alumina (Al2O3) and zirconia (ZrO2), silicon nitride (Si3N4) has been investigated as a novel bioceramic material, mainly in situations where a bone replacement is required. Si3N4 ceramics and its derivative form, SiAlON, possess advantages in orthopedics due to their mechanical properties and biologically acceptable chemistry, which accelerates bone repair. However, biological applications require additional properties, enabling stronger chemical bonding to the surrounding tissue for better fixation and the prevention of bacteria biofilm formation. Therefore, two commercial Si3N4 and SiAlON ceramics were investigated in this study and compared to each other according to their material properties (like wetting angles and surface chemistry) and their antibacterial behaviors using E. coli. Results provided evidence of a 15% reduction in E. coli colonization after just 24 h on Si3N4 compared to SiAlON which is impressive considering no antibiotics were used. Further, a mechanism of action is provided. In this manner, this study provides evidence that Si3N4 should be further studied for a wide range of antibacterial orthopedic, or other suitable biomaterial applications.
Citation - WoS: 17
Citation - Scopus: 19
A comprehensive analysis of the stochastic fractal–fractional tuberculosis model via Mittag-Leffler kernel and white noise
(Elsevier, 2022) Rashid, Saima; Jarad, Fahd; Iqbal, Muhammad Kashif; Alshehri, Ahmed M.; Ashraf, Rehana; Jarad, Fahd; 234808; Matematik
In this research, we develop a stochastic framework for analysing tuberculosis (TB) evolution that includes new-born immunization via the fractal-fractional (F-F) derivative in the Atangana-Baleanu sense. The population is divided into four groups by this system: susceptibility S(xi), infectious I(xi), immunized infants V(xi), and restored R(xi). The stochastic technique is used to describe and assess the invariant region, basic reproduction number, and local stability for disease-free equilibrium. This strategy has significant modelling difficulties since it ignores the unpredictability of the system phenomena. To prevent such problems, we convert the deterministic strategy to a randomized one, which seems recognized to have a vital influence by adding an element of authenticity and fractional approach. Owing to the model intricacies, we established the existence-uniqueness of the model and the extinction of infection was carried out. We conducted a number of experimental tests using the F-F derivative approach and obtained some intriguing modelling findings in terms of (i) varying fractional-order (phi) and fixing fractal-dimension (omega), (ii) varying omega and fixing phi, and (iii) varying both phi and omega, indicating that a combination of such a scheme can enhance infant vaccination and adequate intervention of infectious patients can give a significant boost.
Citation - WoS: 11
Citation - Scopus: 11
A Computational Approach Based On The Fractional Euler Functions And Chebyshev Cardinal Functions For Distributed-Order Time Fractional 2D Diffusion Equation
(Elsevier, 2023) Heydari, M. H.; Hosseininia, M.; Baleanu, D.; 56389
In this paper, the distributed-order time fractional diffusion equation is introduced and studied. The Caputo fractional derivative is utilized to define this distributed-order fractional derivative. A hybrid approach based on the fractional Euler functions and 2D Chebyshev cardinal functions is proposed to derive a numerical solution for the problem under consideration. It should be noted that the Chebyshev cardinal functions process many useful properties, such as orthogonal-ity, cardinality and spectral accuracy. To construct the hybrid method, fractional derivative oper-ational matrix of the fractional Euler functions and partial derivatives operational matrices of the 2D Chebyshev cardinal functions are obtained. Using the obtained operational matrices and the Gauss-Legendre quadrature formula as well as the collocation approach, an algebraic system of equations is derived instead of the main problem that can be solved easily. The accuracy of the approach is tested numerically by solving three examples. The reported results confirm that the established hybrid scheme is highly accurate in providing acceptable results.(c) 2023 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/ 4.0/).
Citation - WoS: 20
Citation - Scopus: 26
A finite difference scheme to solve a fractional order epidemic model of computer virus
(Amer inst Mathematical Sciences-aims, 2023) Jarad, Fahd; Rehman, Muhammad Aziz-ur; Imran, Muhammad; Ahmed, Nauman; Fatima, Umbreen; Akgul, Ali; Jarad, Fahd; Matematik
In this article, an analytical and numerical analysis of a computer virus epidemic model is presented. To more thoroughly examine the dynamics of the virus, the classical model is transformed into a fractional order model. The Caputo differential operator is applied to achieve this. The Jacobian approach is employed to investigate the model's stability. To investigate the model's numerical solution, a hybridized numerical scheme called the Grunwald Letnikov nonstandard finite difference (GL-NSFD) scheme is created. Some essential characteristics of the population model are scrutinized, including positivity boundedness and scheme stability. The aforementioned features are validated using test cases and computer simulations. The mathematical graphs are all detailed. It is also investigated how the fundamental reproduction number R0 functions in stability analysis and illness dynamics.
Citation - WoS: 24
Citation - Scopus: 29
A FRACTAL FRACTIONAL MODEL FOR CERVICAL CANCER DUE TO HUMAN PAPILLOMAVIRUS INFECTION
(World Scientific Publ Co Pte Ltd, 2021) Akgul, A.; Baleanu, Dumitru; Ahmed, N.; Raza, A.; Iqbal, Z.; Rafiq, M.; Rehman, M. A.; Baleanu, D.; 56389; Matematik
In this paper, we have investigated women's malignant disease, cervical cancer, by constructing the compartmental model. An extended fractal-fractional model is used to study the disease dynamics. The points of equilibria are computed analytically and verified by numerical simulations. The key role of R-0 in describing the stability of the model is presented. The sensitivity analysis of R-0 for deciding the role of certain parameters altering the disease dynamics is carried out. The numerical simulations of the proposed numerical technique are demonstrated to test the claimed facts.
Citation - WoS: 35
Citation - Scopus: 44
A fractional derivative with non-singular kernel for interval-valued functions under uncertainty
(Elsevier Gmbh, Urban & Fischer verlag, 2017) Salahshour, S.; Baleanu, Dumitru; Ahmadian, A.; Ismail, F.; Baleanu, D.; 56389; Matematik
The purpose of the current investigation is to generalize the concept of fractional derivative in the sense of Caputo Fabrizio derivative (CF-derivative) for interval-valued function under uncertainty. The reason to choose this new approach is originated from the non singularity property of the kernel that is critical to interpret the memory aftermath of the system, which was not precisely illustrated in the previous definitions. We study the properties of CF-derivative for interval-valued functions under generalized Hukuhara-differentiability. Then, the fractional differential equations under this notion are presented in details. We also study three real-world systems such as the falling body problem, Basset and Decay problem under interval-valued CF-differentiability. Our cases involve a demonstration that this new notion is accurately applicable for the mechanical and viscoelastic models based on the interval CF-derivative equations. (C) 2016 Elsevier GmbH. All rights reserved.
Citation - WoS: 51
Citation - Scopus: 55
A fractional model of vertical transmission and cure of vector-borne diseases pertaining to the Atangana-Baleanu fractional derivatives
(Pergamon-elsevier Science Ltd, 2018) Aliyu, Aliyu Isa; Baleanu, Dumitru; Inc, Mustafa; Yusuf, Abdullahi; Baleanu, Dumitru; 56389; Matematik
The model of transmission dynamics of vector-borne diseases with vertical transmission and cure within a targeted population is extended to the concept of fractional differentiation and integration with non-local and non-singular fading memory introduced. The effect of vertical transmission and cure rate on the basic reproduction number is shown. The Atangana-Baleanu fractional operator in caputo sense (ABC) with non-singular and non-local kernels is used to study the model. The existence and uniqueness of solutions are investigated using the Picard-Lindel method. Ultimately, for illustrating the acquired results, we perform some numerical simulations and show graphically to observe the impact of the arbitrary order derivative. It is expected that the proposed model will show better approximation than the classical model established before. (C) 2018 Elsevier Ltd. All rights reserved.
Citation - WoS: 115
Citation - Scopus: 135
A general fractional formulation and tracking control for immunogenic tumor dynamics
(Wiley, 2022) Jajarmi, Amin; Baleanu, Dumitru; Baleanu, Dumitru; Vahid, Kianoush Zarghami; Mobayen, Saleh; 56389; Matematik
Mathematical modeling of biological systems is an important issue having significant effect on human beings. In this direction, the description of immune systems is an attractive topic as a result of its ability to detect and eradicate abnormal cells. Therefore, this manuscript aims to investigate the asymptotic behavior of immunogenic tumor dynamics based on a new fractional model constructed by the concept of general fractional operators. We discuss the stability and equilibrium points corresponding to the new model; then we modify the predictor-corrector method in general sense to implement the model and compare the associated numerical results with some real experimental data. As an achievement, the new model provides a degree of flexibility enabling us to adjust the complex dynamics of biological system under study. Consequently, the new general model and its solution method presented in this paper for the immunogenic tumor dynamics are new and comprise quite different information than the other kinds of classical and fractional equations. In addition to these, we implement a tracking control method in order to decrease the development of tumor-cell population. The satisfaction of control purpose is confirmed by some simulation results since the controlled variables track the tumor-free steady state in the whole realistic cases.
Citation - WoS: 79
Citation - Scopus: 82
A generalized contraction principle with control functions on partial metric spaces
(Pergamon-elsevier Science Ltd, 2012) Abdeljawad, Thabet; Abdeljawad, Thabet; Karapınar, Erdal; Karapinar, Erdal; Tas, Kenan; Taş, Kenan; 19184; 4971; Matematik
Partial metric spaces were introduced by Matthews in 1994 as a part of the study of denotational semantics of data flow networks. In this article, we prove a generalized contraction principle with control functions phi and psi on partial metric spaces. The theorems we prove generalize many previously obtained results. We also give some examples showing that our theorems are indeed proper extensions. (C) 2011 Elsevier Ltd. All rights reserved.
Citation - WoS: 10
Citation - Scopus: 11
A generalized operational matrix of mixed partial derivative terms with applications to multi-order fractional partial differential equations
(Elsevier, 2022) Talib, Imran; Jarad, Fahd; Jarad, Fahd; Mirza, Muhammad Umar; Nawaz, Asma; Riaz, Muhammad Bilal; 234808; Matematik
In this paper, a computational approach based on the operational matrices in conjunction with orthogonal shifted Legendre polynomials (OSLPs) is designed to solve numerically the multi-order partial differential equations of fractional order consisting of mixed partial derivative terms. Our computational approach has ability to reduce the fractional problems into a system of Sylvester types matrix equations which can be solved by using MATLAB builtin function lyap (.). The solution is approximated as a basis vectors of OSLPs. The efficiency and the numerical stability is examined by taking various test examples. (C) 2021 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Alexandria University.
Citation - WoS: 1
Citation - Scopus: 1
A Higher-Order Approach For Time-Fractional Generalized Burgers' Equation
(World Scientific Publ Co Pte Ltd, 2023) Taneja, Komal; Baleanu, Dumitru; Deswal, Komal; Kumar, Devendra; Baleanu, Dumitru; 56389; Matematik
A fast higher-order scheme is established for solving inhomogeneous time-fractional generalized Burgers' equation. The time-fractional operator is taken as the modified operator with the Mittag-Leffler kernel. Through stability analysis, it has been demonstrated that the proposed numerical approach is unconditionally stable. The convergence of the numerical method is analyzed theoretically using von Neumann's method. It has been proved that the proposed numerical method is fourth-order convergent in space and second-order convergent in time in the L-2-norm. The scheme's proficiency and effectiveness are examined through two numerical experiments to validate the theoretical estimates. The tabular and graphical representations of numerical results confirm the high accuracy and versatility of the scheme.
Citation - WoS: 19
Citation - Scopus: 19
A highly accurate Jacobi collocation algorithm for systems of high-order linear differential-difference equations with mixed initial conditions
(Wiley, 2015) Bhrawy, A. H.; Baleanu, Dumitru; Doha, E. H.; Baleanu, D.; Hafez, R. M.; 56389; Matematik
In this paper, a shifted Jacobi-Gauss collocation spectral algorithm is developed for solving numerically systems of high-order linear retarded and advanced differential-difference equations with variable coefficients subject to mixed initial conditions. The spatial collocation approximation is based upon the use of shifted Jacobi-Gauss interpolation nodes as collocation nodes. The system of differential-difference equations is reduced to a system of algebraic equations in the unknown expansion coefficients of the sought-for spectral approximations. The convergence is discussed graphically. The proposed method has an exponential convergence rate. The validity and effectiveness of the method are demonstrated by solving several numerical examples. Numerical examples are presented in the form of tables and graphs to make comparisons with the results obtained by other methods and with the exact solutions more easier. Copyright (C) 2015 John Wiley & Sons, Ltd.
Citation - WoS: 13
Citation - Scopus: 27
A hybrid analytical algorithm for thin film flow problem occurring in non-Newtonian fluid mechanics
(Elsevier, 2021) Sushila; Baleanu, Dumitru; Singh, Jagdev; Kumar, Devendra; Baleanu, Dumitru; 56389; Matematik
In this work, we investigate thin film flow of a third grade fluid down a inclined plane. The solution of a nonlinear boundary value problem (BVP) is derived by using an effective well organized computational scheme namely homotopy perturbation Elzaki transform method. Furthermore, this model is also resolved by Elzaki decomposition technique. The outcomes achieved by these two approaches are consistent with each other and because of that this technique may be regarded as an optional and effective scheme for determining results of linear and nonlinear BVP. Moreover, the homotopy perturbation Elzaki transform method leads over the Elzaki decomposition method since the nonlinear problems are solved without utilization of Adomian polynomials. (C) 2020 The Authors. Published by Elsevier B.V. on behalf of Faculty of Engineering, Ain Shams University.
Citation - WoS: 19
Citation - Scopus: 24
A hybrid analytical technique for solving nonlinear fractional order PDEs of power law kernel: Application to KdV and Fornberg-Witham equations
(Amer inst Mathematical Sciences-aims, 2022) Ahmad, Shabir; Jarad, Fahd; Ullah, Aman; Akgul, Ali; Jarad, Fahd; 234808; Matematik
It is important to deal with the exact solution of nonlinear PDEs of non-integer orders. Integral transforms play a vital role in solving differential equations of integer and fractional orders. 'o obtain analytical solutions to integer and fractional-order DEs, a few transforms, such as Laplace transforms, Sumudu transforms, and Elzaki transforms, have been widely used by researchers. We propose the Yang transform homotopy perturbation (YTHP) technique in this paper. We present the relation of Yang transform (YT) with the Laplace transform. We find a formula for the YT of fractional derivative in Caputo sense. We deduce a procedure for computing the solution of fractional-order nonlinear PDEs involving the power-law kernel. We show the convergence and error estimate of the suggested method. We give some examples to illustrate the novel method. We provide a comparison between the approximate solution and exact solution through tables and graphs.
Citation - WoS: 112
Citation - Scopus: 124
A hybrid computational approach for Klein-Gordon equations on Cantor sets
(Springer, 2017) Kumar, Devendra; Baleanu, Dumitru; Singh, Jagdev; Baleanu, Dumitru; Matematik
In this letter, we present a hybrid computational approach established on local fractional Sumudu transform method and homotopy perturbation technique to procure the solution of the Klein-Gordon equations on Cantor sets. Four examples are provided to show the accuracy and coherence of the proposed technique. The outcomes disclose that the present computational approach is very user friendly and efficient to compute the nondifferentiable solution of Klein-Gordon equation involving local fractional operator.
Citation - Scopus: 16
A hybrid fractional COVID-19 model with general population mask use: Numerical treatments
(Elsevier B.V., 2021) Sweilam, N.H.; Baleanu, Dumitru; AL-Mekhlafi, S.M.; Almutairi, A.; Baleanu, D.; 56389; Matematik
In this work, a novel mathematical model of Coronavirus (2019-nCov) with general population mask use with modified parameters. The proposed model consists of fourteen fractional-order nonlinear differential equations. Grünwald-Letnikov approximation is used to approximate the new hybrid fractional operator. Compact finite difference method of six order with a new hybrid fractional operator is developed to study the proposed model. Stability analysis of the used methods are given. Comparative studies with generalized fourth order Runge–Kutta method are given. It is found that, the proposed model can be described well the real data of daily confirmed cases in Egypt. © 2021 THE AUTHORS
Citation - WoS: 36
Citation - Scopus: 36
A hybrid fractional optimal control for a novel Coronavirus (2019-nCov) mathematical model
(Elsevier, 2021) Sweilam, N. H.; Baleanu, Dumitru; AL-Mekhlafi, S. M.; Baleanu, D.; 56389; Matematik
Introduction: Coronavirus COVID-19 pandemic is the defining global health crisis of our time and the greatest challenge we have faced since world war two. To describe this disease mathematically, we noted that COVID-19, due to uncertainties associated to the pandemic, ordinal derivatives and their associated integral operators show deficient. The fractional order differential equations models seem more consistent with this disease than the integer order models. This is due to the fact that fractional derivatives and integrals enable the description of the memory and hereditary properties inherent in various materials and processes. Hence there is a growing need to study and use the fractional order differential equations. Also, optimal control theory is very important topic to control the variables in mathematical models of infectious disease. Moreover, a hybrid fractional operator which may be expressed as a linear combination of the Caputo fractional derivative and the Riemann-Liouville fractional integral is recently introduced. This new operator is more general than the operator of Caputo's fractional derivative. Numerical techniques are very important tool in this area of research because most fractional order problems do not have exact analytic solutions. Objectives: A novel fractional order Coronavirus (2019-nCov) mathematical model with modified parameters will be presented. Optimal control of the suggested model is the main objective of this work. Three control variables are presented in this model to minimize the number of infected populations. Necessary control conditions will be derived. Methods: The numerical methods used to study the fractional optimality system are the weighted average nonstandard finite difference method and the Grunwald-Letnikov nonstandard finite difference method. Results: The proposed model with a new fractional operator is presented. We have successfully applied a kind of Pontryagin's maximum principle and were able to reduce the number of infected people using the proposed numerical methods. The weighted average nonstandard finite difference method with the new operator derivative has the best results than Grunwald-Letnikov nonstandard finite difference method with the same operator. Moreover, the proposed methods with the new operator have the best results than the proposed methods with Caputo operator. Conclusions: The combination of fractional order derivative and optimal control in the Coronavirus (2019-nCov) mathematical model improves the dynamics of the model. The new operator is more general and suitable to study the optimal control of the proposed model than the Caputo operator and could be more useful for the researchers and scientists. (C) 2021 The Authors. Published by Elsevier B.V. on behalf of Cairo University.

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