Yazar "Yildirim, Ahmet" seçeneğine göre listele

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    A novel approach to fractional differential equations via JSN transform coupled fractional residual power series method
    (IOP Publishing Ltd, 2024) Prakash, K. Jayaram; Saratha, S. R.; Yildirim, Ahmet
    In this study, we introduce a novel modified general integral transform known as the JSN transform, which offers several advantages over the Laplace and other integral transforms with exponential kernels. Fundamental results of the JSN transform of the Caputo fractional derivative are discussed. Furthermore, we develop a novel hybrid technique called the JSN Fractional Residual Power Series Method ( JSN FRPSM). This new technique incorporates the JSN transform with the existing Residual Power Series Method. To demonstrate the efficiency of the proposed hybrid technique in solving fractional differential equations, we apply it to various fractional differential equations encountered in science and engineering. Statistical and error analyses are conducted to validate the results obtained through the proposed method. Additionally, the series solutions obtained via the proposed method are illustrated graphically.
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    Adaptive multi-step differential transformation method to solving nonlinear differential equations
    (Pergamon-Elsevier Science Ltd, 2012) Gokdogan, Ahmet; Merdan, Mehmet; Yildirim, Ahmet
    In this paper, we propose a fast and effective adaptive algorithm for the multi-step differential transformation method (MsDTM). This approach named as the adaptive MsDTM is applied to a number of nonlinear differential equations and numerical results are given. At the same time, a comparison between the MsDTM and the adaptive MsDTM reveals that the proposed approach is an efficiency tool for solving the considered equations using fewer time steps. (C) 2011 Elsevier Ltd. All rights reserved.
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    An Algorithm for Solving the Fractional Nonlinear Schrodinger Equation by Means of the Homotopy Perturbation Method
    (Freund Publishing House Ltd, 2009) Yildirim, Ahmet
    In this study, we present a framework to obtain analytical solutions to nonlinear fractional Schrodinger equations. The homotopy perturbation method (HPM) is employed to derive analytical solutions for these equations. Some examples are tested and the results reveal that the technique introduced here is very effective and convenient for solving nonlinear partial differential equations of fractional order.
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    Analysis of Fractional Nonlinear Differential Equations Using the Homotopy Perturbation Method
    (Verlag Z Naturforsch, 2011) Balci, Mehmet Ali; Yildirim, Ahmet
    In this study, we used the homotopy perturbation method (HPM) for solving fractional nonlinear differential equations. Three models with fractional-time derivative of order alpha, 0 < alpha < 1, are considered and solved. The numerical results demonstrate that this method is relatively accurate and easily implemented.
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    Analytical Approach to (2+1)-Dimensional Boussinesq Equation and (3+1)-Dimensional Kadomtsev-Petviashvili Equation
    (Verlag Z Naturforsch, 2010) Sariaydin, Selin; Yildirim, Ahmet
    In this paper, we studied the solitary wave solutions of the (2+1)-dimensional Boussinesq equation u(tt) - u(xx) - u(yy) - (u(2))(xx) - u(xxxx) = 0 and the (3+1)-dimensional Kadomtsev-Petviashvili (KP) equation u(xt) - 6u(x)(2) + 6uu(xx) - u(xxxx) - u(yy) - u(zz) = 0. By using this method, an explicit numerical solution is calculated in the form of a convergent power series with easily computable components. To illustrate the application of this method numerical results are derived by using the calculated components of the homotopy perturbation series. The numerical solutions are compared with the known analytical solutions. Results derived from our method are shown graphically.
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    Analytical Approach to a Slowly Deforming Channel Flow with Weak Permeability
    (Walter De Gruyter Gmbh, 2010) Mohyud-Din, Syed Tauseef; Yildirim, Ahmet; Sezer, Sefa Anil
    In this paper, we develop the analytical solution of the Navier-Stokes equations for a semi-infinite rectangular channel with porous and uniformly expanding or contracting walls by employing the homotopy perturbation method (HPM). The series solution of the governing problem is obtained. Some examples have been included. The results so obtained are compared with the existing literature and a remarkable improvement leads to an excellent agreement with the numerical results.
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    Analytical approach to Boussinesq equation with space- and time-fractional derivatives
    (Wiley, 2011) Yildirim, Ahmet; Sezer, Sefa Anil; Kaplan, Yasemin
    In this paper, the homotopy perturbation method (HPM) is developed to obtain approximate analytical solutions of a fractional Boussinesq equation with initial condition. The fractional derivatives are described in the Caputo sense. Some examples are given and comparisons are made, the comparisons show that the HPM is very effective and convenient and overcomes the difficulty of traditional methods. The numerical results show that the approaches are easy to implement and accurate when applied to space- and time-fractional equations. Copyright (C) 2010 John Wiley & Sons, Ltd.
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    Analytical Approach to Fractional Zakharov-Kuznetsov Equations by He's Homotopy Perturbation Method
    (Iop Publishing Ltd, 2010) Yildirim, Ahmet; Gulkanat, Yagmur
    The aim of this paper is to obtain the approximate analytical solution of a fractional Zakharov-Kuznetsov equation by using homotopy perturbation method (HPM). The fractional derivatives are described in the Caputosense. Several examples are given and the results are compared to exact solutions. The results reveal that the method is very effective and simple.
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    Analytical Approach to Space- and Time-Fractional Burgers Equations
    (Iop Publishing Ltd, 2010) Yildirim, Ahmet; Mohyud-Din, Syed Tauseef
    A scheme is developed to study numerical solution of the space- and time-fractional Burgers equations under initial conditions by the homotopy analysis method. The fractional derivatives are considered in the Caputo sense. The solutions are given in the form of series with easily computable terms. Numerical solutions are calculated for the fractional Burgers equation to show the nature of solution as the fractional derivative parameter is changed.
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    AN ANALYTICAL APPROACH TO TRANSMISSION DYNAMICS OF INFECTIOUS DISEASES WITH WANING IMMUNITY
    (World Scientific Publ Co Pte Ltd, 2011) Yildirim, Ahmet; Kocak, Hueseyin
    In this study, we present a mathematical model that describes the dynamics of reinfection under the assumption that the vaccine-induced immune protection may wane over time. The qualitative analysis reveals that the disease eradication depends on vaccination coverage as well as on vaccine efficacy. Using an appropriate Lyapunov function, we established that the disease-free equilibrium is globally asymptotically stable if the vaccination coverage level exceeds a certain threshold value. We combined homotopy perturbation method (HPM) and Pade approximation technique for this problem. Numerical simulations support our analytical conclusions and illustrate possible behavior scenarios of the model.
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    Analytical approximate solution of a SIR epidemic model with constant vaccination strategy by homotopy perturbation method
    (Emerald Group Publishing Limited, 2009) Yildirim, Ahmet; Cherruault, Yves
    Purpose - The purpose of this paper is to introduce an efficient method for solving susceptible-infected-removed (SIR) epidemic model. A SIR model that monitors the temporal dynamics of a childhood disease in the presence of preventive vaccine. The qualitative analysis reveals the vaccination reproductive number for disease control and eradication. It introduces homotopy perturbation method (HPM) to overcome these problems. Design/methodology/approach - The paper considers HPM to solve differential system which describes SIR epidemic model. The essential idea of this method is to introduce a homotopy parameter, say p, which takes values from 0 to 1. When p = 0, the system of equations usually reduces to a sufficiently simplified form, which normally admits a rather simple solution. As p is gradually increased to 1, the system goes through a sequence of deformations, the solution for each of which is close to that at the previous stage of deformation. One of the most remarkable features of the HPM is that usually just few perturbation terms are sufficient for obtaining a reasonably accurate solution. Findings - HPM is employed to compute an approximation to the solution of the non-linear system of differential equations governing the problem. Originality/value - The paper is of value in presenting, via some tables and figures, some numerical experiments which resulted from applying new methods on test problem.
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    Analytical solution for nonlinear wave propagation in shallow media using the variational iteration method
    (Taylor & Francis Ltd, 2012) Younesian, Davood; Askari, Hassan; Saadatnia, Zia; Yildirim, Ahmet
    An analytical solution is presented for nonlinear surface wave propagation. A variational iteration method (VIM) was employed and time-dependent profiles of the surface elevation level and velocity obtained analytically for different initial conditions. It is shown that the VIM used here is a flexible and accurate approach and that it can rapidly converge to the same results obtained by the Adomian decomposition method.
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    Analytical Solution for the Mass Transfer of Ozone of the Second Order from Gaseous Phase to Aqueous Phase
    (Asian Journal Of Chemistry, 2011) Yildirim, Ahmet; Balci, Mehmet Ali
    In this work, we consider the problem of mass transfer of ozone of the second order from a gaseous phase into an aqueous phase by employing the homotopy perturbation method. The series solution of the governing system of differential equations is obtained. Some examples have been included. The effects of the temperature and hydroxyl ion reaction order to the solutions are illustrated by some plots.
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    ANALYTICAL SOLUTION OF MHD STAGNATION-POINT FLOW IN POROUS MEDIA BY MEANS OF THE HOMOTOPY PERTURBATION METHOD
    (Begell House Inc, 2012) Yildirim, Ahmet; Sezer, Sefa Anil
    In this study, the steady two-dimensional laminar forced magnetohydrodynamic Hiemenz flow against a flat plate with variable wall temperature in a porous medium is solved analytically by using the homotopy perturbation method (HPM). The nonlinear boundary layer equations were transformed, and the resulting ordinary differential equations were solved by HPM. The skin friction coefficient and the rate of heat transfer given by the HPM are in good agreement with the numerical solutions of the Keller box method.
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    Analytical Solution of Second-Order Hyperbolic Telegraph Equation by Variational Iteration and Homotopy Perturbation Methods
    (Springer Basel Ag, 2012) Raftari, Behrouz; Yildirim, Ahmet
    In this research, two analytical methods, namely homotopy perturbation method and variational iteration method are introduced to obtain solutions of the initial value problem of hyperbolic type which is called telegraph equation. Some illustrative examples are presented to show the efficiency of the methods.
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    Analytical solution of wave system in R-n with coupling controllers
    (Emerald Group Publishing Ltd, 2011) Mohyud-Din, Syed Tauseef; Yildirim, Ahmet; Demirli, Gulseren
    Purpose - The purpose of this paper is to consider analytical solution of wave system in R-n with coupling controllers by using the homotopy perturbation method (HPM). Design/methodology/approach - HPM is applied to the system of linear partial differential equations, i.e. the system of waves in the two-dimensional version of system equations (1) and (2). This problem is motivated by an analogous problem in ordinary differential equations for coupled oscillators and has potential application in isolating a vibrating object from the outside disturbances. For example, rubber or rubber-like materials can be used to either absorb or shield a structure from vibration. As an approximation, these materials can be modeled as distributed springs. Findings - In this paper, HPM was used to obtain analytical solution of wave system in with coupling controllers. The method provides the solutions in the form of a series with easily computable terms. Unlike other common methods for solving any physical problem, linear or nonlinear, that requires linearization, discretization, perturbation, or unjustified assumptions that may slightly change the physics of the problem, the HPM finds approximate analytical solutions by using the initial conditions only. Originality/value - The method proposed in this paper is very reliable and efficient and is being used quite extensively for diversified nonlinear problems of a physical nature. The algorithm is being used for the first time on such problems.
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    Analytical Solutions for Autonomous Conservative Nonlinear Oscillator
    (Walter De Gruyter Gmbh, 2010) Yazdi, Mohammad Kalami; Khan, Yasir; Madani, M.; Askari, H.; Saadatnia, Z.; Yildirim, Ahmet
    This paper adapts the Energy Balance Method (EBM) and Frequency Amplitude Formulation to solve the free vibrations of a conservative oscillator with inertia and static cubic non-linearities. Case studies on the effects of the time response are presented. The results that obtained from the EBM and FAF are then compared with those from the numerical solution in order to verify the accuracy of the proposed method.
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    Application of Differential Transformation Method for Numerical Computation of Regularized Long Wave Equation
    (Verlag Z Naturforsch, 2012) Soltanalizadeh, Babak; Yildirim, Ahmet
    In this article, the differential transformation method (DTM) is utilized for finding the solution of the regularized long wave (RLW) equation. Not only the exact solutions have been achieved by the known forms of the series solutions, but also for the finite terms of series, and the corresponding numerical approximations have been computed.
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    Application of He's Variational Iteration Method to Nonlinear Integro-Differential Equations
    (Verlag Z Naturforsch, 2010) Yildirim, Ahmet
    In this paper, an application of He's variational iteration method is applied to solve nonlinear integro-differential equations. Some examples are given to illustrate the effectiveness of the method. The results show that the method provides a straightforward and powerful mathematical tool for solving various nonlinear integro-differential equations.
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    APPLICATION OF HIGHER ORDER HAMILTONIAN APPROACH TO NONLINEAR VIBRATING SYSTEMS
    (Polish Soc Theoretical & Applied Mechanics, 2013) Askari, Hassan; Nia, Zia Saadat; Yildirim, Ahmet; Yazdi, Mohammed K.; Khan, Yasir
    The higher order Hamiltonian approach is utilized to elicit approximate solutions for two nonlinear oscillation systems. Frequency-amplitude relationships and the model of buckling of a column and mass-spring system are scrutinized in this paper. First, second and third approximate solutions of examples are achieved, and the frequency responses of the systems are verified by exact numerical solutions. According to the numerical results, we can conclude that the Hamiltonian approach is an applicable method for solving the nonlinear equations, and the accuracy of this method in the second and third approximates is very high and reliable. The achieved results of this paper demonstrate that this method is powerful and uncomplicated for solving of sophisticated nonlinear problems.