Matematik Bölümü

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A Brief Overview and Survey of the Scientific Work by Feng Qi
(2022) Agarwal, Ravi Prakash; Karapinar, Erdal; Kostić, Marko; Cao, Jian; Du, Wei-Shih; 19184
In the paper, the authors present a brief overview and survey of the scientific work by Chinese mathematician Feng Qi and his coauthors.
A close look at Newton–Cotes integration rules
(2019) Sermutlu, Emre; 17647
Newton–Cotes integration rules are the simplest methods in numerical integration. The main advantage of using these rules in quadrature software is ease of programming. In practice, only the lower orders are implemented or tested, because of the negative coefficients of higher orders. Most textbooks state it is not necessary to go beyond Boole’s 5-point rule. Explicit coefficients and error terms for higher orders are seldom given literature. Higher-order rules include negative coefficients therefore roundoff error increases while truncation error decreases as we increase the number of points. But is the optimal one really Simpson or Boole? In this paper, we list coefficients up to 19-points for both open and closed rules, derive the error terms using an elementary and intuitive method, and test the rules on a battery of functions to find the optimum all-round one.
A common fixed point theorem of a Greguš type on convex cone metric spaces
(2011) Abdeljawad, Thabet; Karapinar, Erdal; 19184
The result of Ćirić [1] on a common fixed point theorem of Greguš type on metric spaces is extended to the class of cone metric spaces. Namely, a common fixed point theorem is proved in s-convex cone metric spaces under the normality of the cone and another common fixed point theorem is proved in convex cone metric spaces under the assumption that the cone is strongly minihedral.
A comprehensive analysis of the stochastic fractal–fractional tuberculosis model via Mittag-Leffler kernel and white noise
(2022) Jarad, Fahd; Iqbal, Muhammad Kashif; Alshehri, Ahmed M.; Ashraf, Rehana; Jarad, Fahd; 234808
In this research, we develop a stochastic framework for analysing tuberculosis (TB) evolution that includes newborn 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(ξ), infectious I(ξ), immunized infants V(ξ), and restored R(ξ). 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 (φ) and fixing fractal-dimension (ω), (ii) varying ω and fixing φ, and (iii) varying both φ and ω, indicating that a combination of such a scheme can enhance infant vaccination and adequate intervention of infectious patients can give a significant boost.
A computational study of a stochastic fractal-fractional hepatitis B virus infection incorporating delayed immune reactions via the exponential decay
(2022) Jarad, Fahd; Rashid, Saima; Jarad, Fahd; 234808
Recently, researchers have become interested in modelling, monitoring, and treatment of hepatitis B virus infection. Understanding the various connections between pathogens, immune systems, and general liver function is crucial. In this study, we propose a higher-order stochastically modified delay differential model for the evolution of hepatitis B virus transmission involving defensive cells. Taking into account environmental stimuli and ambiguities, we presented numerical solutions of the fractal-fractional hepatitis B virus model based on the exponential decay kernel that reviewed the hepatitis B virus immune system involving cytotoxic T lymphocyte immunological mechanisms. Furthermore, qualitative aspects of the system are analyzed such as the existence-uniqueness of the non-negative solution, where the infection endures stochastically as a result of the solution evolving within the predetermined system’s equilibrium state. In certain settings, infection-free can be determined, where the illness settles down tremendously with unit probability. To predict the viability of the fractal-fractional derivative outcomes, a novel numerical approach is used, resulting in several remarkable modelling results, including a change in fractional-order δ with constant fractal-dimension $, δ with changing $, and δ with changing both δ and $. White noise concentration has a significant impact on how bacterial infections are treated.
A Decomposıtıon Algorıthm Coupled Wıth Operatıonal Matrıces Approach Wıth Applıcatıons To Fractıonal Dıfferentıal Equatıons
(2021) Baleanu, Dumitru; Alam, Md. Nur; Baleanu, Dumitru; Zaidi, Danish; 56389
In this article, we solve numerically the linear and non-linear fractional initial value problems of multiple orders by developing a numerical method that is based on the decomposition algorithm coupled with the operational matrices approach. By means of this, the fractional initial value problems of multiple orders are decomposed into a system of fractional initial value problems which are then solved by using the operational matrices approach. The efficiency and advantage of the developed numerical method are highlighted by comparing the results obtained otherwise in the literature. The construction of the new derivative operational matrix of fractional legendre function vectors in the Caputo sense is also a part of this research. As applications, we solve several fractional initial value problems of multiple orders. The numerical results are displayed in tables and plots.
A delayed plant disease model with Caputo fractional derivatives
(2022) Baleanu, Dumitru; Baleanu, Dumitru; Erturk, Vedat Suat; Inc, Mustafa; Govindaraj, V.; 56389
We analyze a time-delay Caputo-type fractional mathematical model containing the infection rate of Beddington–DeAngelis functional response to study the structure of a vector-borne plant epidemic. We prove the unique global solution existence for the given delay mathematical model by using fixed point results. We use the Adams–Bashforth–Moulton P-C algorithm for solving the given dynamical model. We give a number of graphical interpretations of the proposed solution. A number of novel results are demonstrated from the given practical and theoretical observations. By using 3-D plots we observe the variations in the flatness of our plots when the fractional order varies. The role of time delay on the proposed plant disease dynamics and the effects of infection rate in the population of susceptible and infectious classes are investigated. The main motivation of this research study is examining the dynamics of the vector-borne epidemic in the sense of fractional derivatives under memory effects. This study is an example of how the fractional derivatives are useful in plant epidemiology. The application of Caputo derivative with equal dimensionality includes the memory in the model, which is the main novelty of this study.
A Detailed Study on a Tumor Model with Delayed Growth of Pro-Tumor Macrophages
(2022) Dehingia, Kaushik; Hosseini, Kamyar; Salahshour, Soheil; Baleanu, D.; 56389
This paper investigates a tumor-macrophages interaction model with a discrete-time delay in the growth of pro-tumor M2 macrophages. The steady-state analysis of the governing model is performed around the tumor dominant steady-state and the interior steady-state. It is found that the tumor dominant steady-state is locally asymptotically stable under certain conditions, and the stability of the interior steady-state is affected by the discrete-time delay; as a result, the unstable system experiences a Hopf bifurcation and gets stabilized. Furthermore, the transversality conditions for the existence of Hopf bifurcations are derived. Several graphical representations in two and three-dimensional postures are given to examine the validity of the results provided in the current study.
A discussion on a pata type contraction via iterate at a point
(2020) Karapınar, Erdal; Fulga, Andreea; Rakočević, Vladimir; 19184
In this paper, we introduce the notion of Pata type contraction at a point in the context of a complete metric space. We observe that such contractions possesses unique fixed point without continuity assumption on the given mapping. Thus, is extended the original results of Pata. We also provide an example to illustrate its validity.
A discussion on the coincidence quasi-best proximity points
(2021) Karapınar, Erdal; Abkar, Ali; Karapınar, Erdal; 19184
In this paper, we first introduce a new class of the pointwise cyclic-noncyclic proximal contraction pairs. Then we consider the coincidence quasi-best proximity point problem for this class. Finally, we study the coincidence quasi-best proximity points of weak cyclic-noncyclic Kannan contraction pairs. We consider an example to indicate the validity of the main result.
A discussion on the existence of positive solutions of the boundary value problems via ψ-Hilfer fractional derivative on b-metric spaces
(2020) Karapınar, Erdal; Karapınar, Erdal; 19184
In this paper, we investigate the existence of positive solutions for the new class of boundary value problems via ψ-Hilfer fractional differential equations. For our purpose, we use the α− ψ Geraghty-type contraction in the framework of the b-metric space. We give an example illustrating the validity of the proved results.
A general treatment of singular Lagrangians with linear velocities
(Editrice Copmpositori Bologna, 2000) Baleanu, Dumitru; Güler, Y.; 56389
The Hamilton-Jacobi treatment of singular systems with linear velocities is investigated. Since the rank of Hessian matrix is zero, all the generalized coordinates are independent parameters. Integrability conditions reduce the degrees of freedom. Path integral quantization is analyzed.
A highly accurate Jacobi collocation algorithm for systems of high-order linear differential-difference equations with mixed initial conditions
(Wiley, 2015) Baleanu, Dumitru; Doha, E. H.; Baleanu, Dumitru; Hafez, R. M.; 56389
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.
A hybrid analytical technique for solving nonlinear fractional order PDEs of power law kernel: Application to KdV and Fornberg-Witham equations
(2022) Jarad, Fahd; Ullah, Aman; Akgül, Ali; Jarad, Fahd; 234808
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. To 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.
A hybrid computing approach to design the novel second order singular perturbed delay differential Lane-Emden model
(2022) Baleanu, Dumitru; Baleanu, Dumitru; Raja, Muhammad Asif Zahoor; Hincal, Evren; 56389
In this study, the mathematical form of the second order perturbed singular delay differential system is presented. The comprehensive features using the singular-point, perturbed factor and pantograph term are provided together with the shape factor of the second order perturbed singular delay differential system. The novel model is simulated numerically through the artificial neural networks (ANNs) based on the global/local optimization procedures, i.e., genetic algorithm (GA) and sequential quadratic programming (SQP). An activation function is constructed by using the differential model based on the second order perturbed singular delay differential system. The optimization of fitness function is performed through the hybrid computing strength of the ANNs-GA-SQP to solve the second order perturbed singular delay differential system. The exactness, substantiation, and authentication of the novel system is observed to solve three different variants of the novel model. The convergence, robustness, correctness, and stability of the numerical approach is performed using the comparison procedures of the available exact solutions. For the reliability, the statistical performances with necessary processes are provided using the ANNs-GA-SQP.
A method of inversion of Fourier transforms and its applications
(Academic Publications LTD, 2019) Kushpel, Alexander; 279144
The problem of inversion of Fourier transforms is a frequently discussed topic in the theory of PDEs, Stochastic Processes and many other branches of Analysis. We consider here in more details an application of a method proposed in Financial Modeling. As a motivating example consider a frictionless market with no arbitrage opportunities and a constant riskless interest rate r > 0. Assuming the existence of a risk-neutral equivalent martingale measure Q, we get the option value V = e −rTE Q[ϕ] at time 0 and maturity T > 0, where ϕ is a reward function and the expectation E Q is taken with respect to the equivalent martingale measure Q. Usually, the reward function ϕ has a simple structure. Hence, the main problem is to approximate properly the respective density function and then to approximate E Q [ϕ]. Here we offer an approximant for the density function without proof of any convergence results. These problems will be considered in details in our future publications.
A Modified Algorithm Based on Haar Wavelets for the Numerical Simulation of Interface Models
(2022) Jarad, Fahd; Asif, Muhammad; Haider, Nadeem; Bilal, Rubi; Ahsan, Muhammad; Al-Mdallal, Qasem; Jarad, Fahd; 234808
In this paper, a new numerical technique is proposed for the simulations of advection-diffusion-reaction type elliptic and parabolic interface models. The proposed technique comprises of the Haar wavelet collocation method and the finite difference method. In this technique, the spatial derivative is approximated by truncated Haar wavelet series, while for temporal derivative, the finite difference formula is used. The diffusion coefficients, advection coefficients, and reaction coefficients are considered discontinuously across the fixed interface. The newly established numerical technique is applied to both linear and nonlinear benchmark interface models. In the case of linear interface models, Gauss elimination method is used, whereas for nonlinear interface models, the nonlinearity is removed by using the quasi-Newton linearization technique. The L∞ errors are calculated for different number of collocation points. The obtained numerical results are compared with the immersed interface method. The stability and convergence of the method are also discussed. On the whole, the numerical results show more efficiency, better accuracy, and simpler applicability of the newly developed numerical technique compared to the existing methods in literature.
A modified variational iteration method for solving fractional riccati differential equation by Adomian polynomials
(Walter De Gruyter GMBH, 2013) Baleanu, Dumitru; Tajadodi, Haleh; Baleanu, Dumitru; 56389
In this paper, we introduce a modified variational iteration method (MVIM) for solving Riccati differential equations. Also the fractional Riccati differential equation is solved by variational iteration method with considering Adomians polynomials for nonlinear terms. The main advantage of the MVIM is that it can enlarge the convergence region of iterative approximate solutions. Hence, the solutions obtained using the MVIM give good approximations for a larger interval. The numerical results show that the method is simple and effective.
A New (4 + 1)-Dimensional Burgers Equation: Its Bäcklund Transformation and Real and Complex -Kink Solitons
(2022) Baleanu, Dumitru; Mirzazadeh, Mohammad; Hosseini, Kamyar; Salahshour, Soheil; Baleanu, Dumitru; 56389
Studying the dynamics of solitons in nonlinear evolution equations (NLEEs) has gainedconsiderable interest in the last decades. Accordingly, the search for soliton solutions ofNLEEs has been the main topic of many research studies. In the present paper, a new (4+ 1)-dimensional Burgers equation (n4D-BE) is introduced that describes speciﬁc disper-sive waves in nonlinear sciences. Based on the truncated Painlevé expansion, the Bäcklundtransformation of the n4D-BE is ﬁrstly extracted, then, its real and complex N-kink solitonsare derived using the simpliﬁed Hirota method. Furthermore, several ansatz methods areformally adopted to obtain a group of other single-kink soliton solutions of the n4D-BE
A new analytical method to simulate the mutual impact of space-time memory indices embedded in (1+2)-physical models
(2022) Baleanu, Dumitru; Jaradat, I; Alquran, Marwan; Baleanu, Dumitru; 56389
In the present article, we geometrically and analytically examine the mutual impact of space-time Caputo derivatives embedded in (1 + 2)-physical models. This has been accomplished by integrating the residual power series method (RPSM) with a new trivariate fractional power series representation that encompasses spatial and temporal Caputo derivative parameters. Theoretically, some results regarding the convergence and the error for the proposed adaptation have been established by virtue of the Riemann-Liouville fractional integral. Practically, the embedding of Schrodinger, telegraph, and Burgers' equations into higher fractional space has been considered, and their solutions furnished by means of a rapidly convergent series that has ultimately a closed-form fractional function. The graphical analysis of the obtained solutions has shown that the solutions possess a homotopy mapping characteristic, in a topological sense, to reach the integer case solution where the Caputo derivative parameters behave similarly to the homotopy parameters. Altogether, the proposed technique exhibits a high accuracy and high rate of convergence.

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Browsing Matematik Bölümü by Department "Çankaya Üniversitesi, Fen-Edebiyat Fakültesi, Matematik Bölümü"