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Öğe A FRACTAL FRACTIONAL MODEL FOR CERVICAL CANCER DUE TO HUMAN PAPILLOMAVIRUS INFECTION(World Scientific Publ Co Pte Ltd, 2021) Akgul, A.; Ahmed, N.; Raza, A.; Iqbal, Z.; Rafiq, M.; Rehman, M. A.; Baleanu, D.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.Öğe A New Application Of Reproducing Kernel Hilbert Space Method(Amer Inst Physics, 2018) Akgul, A.; Sakar, M. GiyasWe demonstrate a new application of the reproducing kernel Hilbert space method in this paper. We showed our results by tables and figures.Öğe A Numerical Investigation on Burgers Equation by MOL-GPS Method(Amer Scientific Publishers, 2017) Hashemi, M. S.; Inc, M.; Karatas, E.; Akgul, A.Group preserving scheme for calculating the numerical solutions of the Burgers equation with appropriate boundary and initial conditions is given in this work. Approximate solutions are demonstrated by tables and figures. Numerical results present the efficiency and power of this technique.Öğe An efficient method for the analytical study of linear and nonlinear time-fractional partial differential equations with variable coefficients(Samara State Technical Univ, 2023) Liaqat, M. I.; Akgul, A.; Prosviryakov, E. Yu.The residual power series method is effective for obtaining approximate analytical solutions to fractional-order differential equations. This method, however, requires the derivative to compute the coefficients of terms in a series solution. Other well-known methods, such as the homotopy perturbation, the Adomian decomposition, and the variational iteration methods, need integration. We are all aware of how difficult it is to calculate the fractional derivative and integration of a function. As a result, the use of the methods mentioned above is somewhat constrained. In this research work, approximate and exact analytical solutions to time-fractional partial differential equations with variable coefficients are obtained using the Laplace residual power series method in the sense of the Gerasimov-Caputo fractional derivative. This method helped us overcome the limitations of the various methods. The Laplace residual power series method performs exceptionally well in computing the coefficients of terms in a series solution by applying the straightforward limit principle at infinity, and it is also more effective than various series solution methods due to the avoidance of Adomian and He polynomials to solve nonlinear problems. The relative, recurrence, and absolute errors of the three problems are investigated in order to evaluate the validity of our method. The results show that the proposed method can be a suitable alternative to the various series solution methods when solving time-fractional partial differential equations.Öğe Chemostat Model Analysis Using Various Kernels with Fractional Derivatives(Islamic Azad Univ, Shiraz Branch, 2023) Akgul, A.; Ulgul, E.; Alqahtani, R. T.The investigation involves utilizing a set of three ordinary differential equations to mathematically model the degradation process of a mixture containing phenol and p-cresol within a continuously agitated bioreactor. The primary focus lies in the stability analysis of equilibrium points within this model. Additionally, the research delves into exploring the influence of fractal dimension and fractional order on the model, incorporating fractal-fractional derivatives and employing three distinct types of kernels. To quantify the concentrations of phenol, p-cresol, and biomass, highly effective computational algorithms have been formulated, enhancing the precision and efficiency of data analysis. In conclusion, the proposed methodology's soundness and accuracy are thoroughly scrutinized and affirmed through extensive computational simulations.Öğe Constructing two powerful methods to solve the Thomas-Fermi equation(Springer, 2017) Akgul, A.; Hashemi, M. S.; Inc, M.; Raheem, S. A.We implement the reproducing kernel method and SL(2, R)-shooting method to solve the Thomas-Fermi equation. Powerful techniques are demonstrated by reproducing kernel functions. The reliable numerical approximations to the solution of this equation are calculated by two novel approaches whose results are in good agreement. Numerical results are shown in order to prove the certainty of the techniques.Öğe Efficient spectral collocation method for nonlinear systems of fractional pantograph delay differential equations(Amer Inst Mathematical Sciences-Aims, 2024) Zaky, M. A.; Babatin, M.; Hammad, M.; Akgul, A.; Hendy, A. S.Caputo-Hadamard-type fractional calculus involves the logarithmic function of an arbitrary exponent as its convolutional kernel, which causes challenges in numerical approximations. In this paper, we construct and analyze a spectral collocation approach using mapped Jacobi functions as basis functions and construct an efficient algorithm to solve systems of fractional pantograph delay differential equations involving Caputo-Hadamard fractional derivatives. What we study is the error estimates of the derived method. In addition, we tabulate numerical results to support our theoretical analysis.Öğe Invariant Investigation on the System of Hirota-Satsuma Coupled KdV Equation(Amer Inst Physics, 2018) Hashemi, M. S.; Balmeh, Z.; Akgul, A.; Akgul, E. K.; Baleanu, D.We show how invariant subspace method can be extended to the system time fractional differential equations and construct their exact solutions. Effectiveness of the method has been illustrated by the time fractional Hirota-Satsuma Coupled KdV(HSCKdV) equation.Öğe Lie symmetries, bifurcation analysis, and Jacobi elliptic function solutions to the nonlinear Kodama equation(Elsevier, 2023) Hosseini, K.; Alizadeh, F.; Hincal, E.; Baleanu, D.; Akgul, A.; Hassan, A. M.The present article formally studies the propagation of optical pulses in a nonlinear medium governed by the nonlinear Kodama equation. For this purpose, the Lie symmetries group is first adopted for similarity reduction and constructing some exact solutions of the governing equation. After deriving the dynamical system of the nonlinear Kodama equation, its bifurcation analysis is accomplished using the idea of the planar dynamical system. Through perturbing the resultant dynamical system using a trigonometric function, chaotic characteristics of the governing model are analyzed by serving several two- and three-dimensional phase portraits. A sensitivity analysis of the dynamical system is performed using the Runge-Kutta method to ensure that small changes in initial conditions have little impact on solution stability. Finally, using the technique of the planar dynamical system, a number of Jacobi elliptic function solutions (in special cases, bright and dark solitons) are constructed for the nonlinear Kodama equation. It has been shown that bright and dark solitons can be controlled for their width and height effectively by the achievements of the current paper.Öğe Nonlinear Self-Adjointness and Nonclassical Solutions of a Population Model with Variable Coefficients(Amer Scientific Publishers, 2018) Hashemi, M. S.; Inc, M.; Akgul, A.; Baleanu, D.In this work, the size-structured population model with variable coefficients is considered to construct the exact solutions with nonclassical symmetries in the light of the heir equations. Nonlinear self-adjointness is shown and conservation laws are calculated too. Some scientific theorems have been given in this paper.Öğe Numerical appraisal of the unsteady Casson fluid flow through Finite Element Method (FEM)(Sharif Univ Technology, 2023) Khader, M. M.; Inc, M.; Akgul, A.The current study proposes an efficient numerical method to evaluate the effects of the variable heat flux, viscous dissipation, and slip velocity on the viscous Casson Heat Transfer (CHT), considering the unsteady stretching sheet that took into account the effect of heat generation or absorption. In this respect, Finite Element Method (FEM) was employed to solve the resulting ODEs that described the problem. The effects of the factors governing the HT such as unsteadiness parameter, slip velocity parameter, Casson parameter, local Eckert number, heat generation parameter, and Prandtl number were studied. In addition, the local skin friction coefficient and local Nusselt number on the stretching sheet were also computed. Finally, the obtained solutions confirmed that the proposed procedure could be an easy and efficient tool for finding a solution to such fluid models. (c) 2023 Sharif University of Technology. All rights reserved.Öğe NUMERICAL RECONSTRUCTION OF A SPACE-DEPENDENT SOURCE TERM FOR MULTIDIMENSIONAL SPACE-TIME FRACTIONAL DIFFUSION EQUATIONS(Editura Acad Romane, 2023) Sidi, H. ould; Zaky, M. A.; EL Waled, K.; Akgul, A.; Hendy, A. S.In this paper, we consider the problem of identifying the unknown source function in the time-space fractional diffusion equation from the final obser-vation data. An implicit difference technique is proposed in conjunction with the matrix transfer scheme for approximating the solution of the direct problem. The challenge pertains to an inverse scenario encompassing a nonlocal ill-posed operator. The problem under investigation is formulated as a regularized optimization problem with a least-squares cost function minimization objective. An approximation for the source function is obtained using an iterative non-stationary Tikhonov regularization approach. Three numerical examples are reported to verify the efficiency of the pro-posed schemes.Öğe On Solutions of Fractional Differential Equations(Amer Inst Physics, 2018) Akgul, A.; Sakar, M. GiyasIn this paper, we obtain exact and approximate solutions of differential equations by reproducing kernel Hilbert space method. We demonstrate our solutions by series.Öğe On the MHD boundary layer flow with diffusion and chemical reaction over a porous flat plate with suction/blowing: two reliable methods(Springer, 2021) Hashemi, M. S.; Akgul, A.In this paper, a Lie-group integrator based on GL4(R) and the reproducing kernel functions has been constructed to investigate the flow characteristics in an electrically conducting second-grade fluid over a stretching sheet. Accurate initial values can be achieved when the target equation is matched precisely, and then, we can apply the group preserving scheme (GPS) to get a rather accurate results. On the other hand, the reproducing kernel method (RKM) is successfully applied to the underlying equation with convergence analysis. We show exact and approximate solutions by series in the reproducing kernel space. We use a bounded linear operator in the reproducing kernel space to get the solutions by the reproducing kernel method. Comparison of these two methods demonstrates the power and reliability. Finally, effects of magnetic parameter, viscoelastic parameter, stagnation-point flow, and stretching of the sheet parameters are illustrated.Öğe Propagation of optical pulses in fiber optics modelled by coupled space-time fractional dynamical system(Elsevier, 2023) Nasreen, N.; Lu, D.; Zhang, Z.; Akgul, A.; Younas, U.; Nasreen, S.; Al-Ahmadi, Ameenah N.This article focuses on securing distinct optical solitons to optical fiber with the coupled nonlinear Schro center dot dinger equation. The examined equation is analyzed with the aid of conformable space-time fractional and is known to have a significant role in the propagation of pulses through a two-mode optical fiber and the soliton wavelength division multiplexing. The fractional nonlinear partial differential equations have garnered increased interest since they may be utilized to explain a wide range of complicated physical phenomena and have more dynamic structures of localized wave solutions. New extended direct algebraic method, a relatively recent integration tool, is used to obtain the solutions. The diverse pulses as bright, dark, combo, and singular soliton solutions have been extracted. In addition to aiding in the clarification of fractional nonlinear partial differential equations, the employed method provides previously extracted solutions and extracts new exact solutions. Given the correct parameter values, numerous graph forms are sketched to provide infor-mation on the visual presentation of the obtained findings. The achieved solutions are to be attrac-tive to researchers for understanding the complexity of the considered model. The findings of this research validate the effectiveness of the proposed method for increasing nonlinear dynamicalÖğe Reproducing Kernel Functions and Bounded Linear Operator for Solving Fractional Nonlinear Boundary Value Problems(Amer Inst Physics, 2018) Akgul, A.; Sakar, M. GiyasIn this work, reproducing kernel functions are given for solving fractional nonlinear boundary value problems. The reproducing kernel functions used in this paper are very important functions to get approximate solutions. Additionally. A bounded linear operator has been found in this work.Öğe Stability and control of the complex chaotic financial system with fractional derivatives(Cambridge Scientific Publishers, 2020) Farman, M.; Ahmad, A.; Akgul, A.; Saleem, M.U.; Naeem, M.We represent a nonlinear time-fractional model of the complex chaotic financial system to understand the occurrences of the interest rate, investment demand and price index in this research. The fractional parameter is used to develop the system of complex nonlinear differential equations by using Caputo with fractional derivative. The stabilization of equilibrium is obtained by both theoretical analysis and the simulation result. A certain threshold value of the basic reproduction number with sensitivity analysis has been made. Stability analysis of the model is confirmed by the Lyapunov equation and detail the stability analysis of this strategy together with the uniqueness of the special solutions. The idea of observability and controllability for linearized control is utilized for feedback control. By utilizing the Laplace Adomian Decomposition technique, the solution of the time-fractional model has been obtained. At last, numerical simulations are given for three fractionalorder chaotic systems to check the effectiveness as indicated by the fractional parameter. © 2020. CSP - Cambridge, UK; I&S - Florida, USA. All Rights Reserved.Öğe Variation in electronic and optical responses due to phase transformation of SrZrO3 from cubic to orthorhombic under high pressure: a computational insight(Indian Assoc Cultivation Science, 2022) Rizwan, M.; Farman, M.; Akgul, A.; Usman, Z.; Anam, S.In this paper, phase transition in SrZrO3 under high pressure has been investigated computationally using first principles calculation with ultra-soft pseudo-potential and generalized gradient approximation suggested by Perdew, Burke, and Ernzerhof correlation functional. The point where the enthalpy vs pressure curves of cubic and orthorhombic phases coincide is observed for the evaluation of phase transition pressure. The phase transition from cubic to orthorhombic has been observed at 72.0 GPa pressure. At phase transition, the electronic band gap, total density of states, lattice parameter, volume and optical properties of SrZrO3 are explored. The band gap as well as the lattice parameter and volume reduce during phase transformation from cubic to orthorhombic. The band gap values at 67.0 GPa for cubic and orthorhombic phases are simultaneously 3.418 eV and 4.117 eV whereas at final pressure of 74.0 GPa, these values are 3.368 eV and 4.108 eV, respectively. The calculated and reported values of refractive index, dielectric function and absorption spectra are comparable. The calculated values of static refractive index are 2.1 and 2.2 for cubic and orthorhombic SrZrO3, respectively.
