Browsing by Author "Bildik, N"

Now showing 1 - 20 of 52
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    Exact Solutions of the Time-Fractional Fisher Equation by using Modified Trial Equation Method
    Tandogan, YA; Bildik, N
    In this study, modified trial equation method has been proposed to obtain precise solutions of nonlinear fractional differential equation. Using the modified test equation method, we obtained some new exact solutions of the time fractional nonlinear Fisher equation. The obtained results are classified as a soliton solution, singular solutions, rational function solutions and periodic solutions.
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    Inelastic analysis of pile soil structure interaction
    Küçükarslan, S; Banerjee, PK; Bildik, N
    In this paper, inelastic pile soil structure interaction is analyzed by using a hybrid type of numerical method. Piles and structural elements are modeled as linear finite elements and soil half space is modeled by using boundary elements. Inelastic modeling of soil media is presented by introducing a rational approximation to continuum with nonlinear interface springs along the piles. For this purpose, modified Ozdemir's nonlinear model is implemented and systems of equations are coupled for piles and pile groups at interacting nodes. To verify the proposed algorithm, four experimental results from previously conducted tests under static loads are compared with those obtained from present analysis. (C) 2003 Elsevier Science Ltd. All rights reserved.
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    Legendre wavelet solution of neutral differential equations with proportional delays
    Gümgüm, S; Özdek, DE; Özaltun, G; Bildik, N
    The aim of this paper is to solve neutral differential equations with proportional delays by using Legendre wavelet method. Using orthonormal polynomials is the main advantage of this method since it enables a decrease in the computational cost and runtime. Some examples are displayed to illustrate the efficiency and accuracy of the proposed method. Numerical results are compared with various numerical methods in literature and show that the present method is very effectual in solving neutral differential equations with proportional delays.
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    On the Solutions of Nonlinear Boussinesq Differential Equations
    Bildik, N; Tandogan, YA
    In recent years, many studies upon development of new techniques for solutions ofthese models and creation of mathematical modelsofreal life problems which encountered in many ofthe applied scienceshave been done. New solution functions were tried to obtain by using development methods related to this type nonlinear physical problems. Especially, soliton solutions, singular solutions and other solutions were obtained for these type physical problems. In this study, trial equation method is handled in order to find new exact solutions of non-integrable physical models. This method is applied to nonlinear partial differential equations. From hence, solution functions in elliptic integral form are obtained. Elliptic functions have kinds such as Elliptic-F, Elliptic-E and Elliptic-Pi.
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    Existence and Numerical Solution of the Volterra Fractional Integral Equations of the Second Kind
    Atangana, A; Bildik, N
    This work presents the possible generalization of the Volterra integral equation second kind to the concept of fractional integral. Using the Picard method, we present the existence and the uniqueness of the solution of the generalized integral equation. The numerical solution is obtained via the Simpson 3/8 rule method. The convergence of this scheme is presented together with numerical results.
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    Multi Wave Method for the Generalized form of BBM Equation
    Bildik, N; Tandogan, YA
    In this paper, we apply the multi-wave method to find new multi wave solutions for an important nonlinear physical model. This model is well known as generalized form of Benjamin Bona Mahony (BBM) equation. Using the mathematics software Mathematica, we compute the traveling wave solutions. Then, the multi wave solutions including periodic wave solutions, bright soliton solutions and rational function solutions are obtained by the multi wave method. It is seen that this method is very useful mathematical approach for generalized form of BBM equation.
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    The Use of Fractional Order Derivative to Predict the Groundwater Flow
    Atangana, A; Bildik, N
    The aim of this work was to convert the Thiem and the Theis groundwater flow equation to the time-fractional groundwater flow model. We first derived the analytical solution of the Theim time-fractional groundwater flow equation in terms of the generalized Wright function. We presented some properties of the Laplace-Carson transform. We derived the analytical solution of the Theis time-fractional groundwater flow equation (TFGFE) via the Laplace-Carson transform method. We introduced the generalized exponential integral, as solution of the TFGFE. This solution is in perfect agreement with the data observed from the pumping test performed by the Institute for Groundwater Study on one of its borehole settled on the test site of the University of the Free State. The test consisted of the pumping of the borehole at the constant discharge rate Q and monitoring the piezometric head for 350 minutes.
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    NUMERICAL SOLUTION OF VOLTERRA SERIES WITH ERROR ESTIMATION
    Tari, A; Bildik, N
    In this paper, an important applied problem, namely Volterra series, is investigated. Here, the well-known differential transform (DT) method is extended to solve the multiple non-linear Volterra integral equations which lead to Volterra series. To this end, some basic properties of DT are given, then by proving some theorems, the DT method is extended to solve the mentioned equations. Next, a scheme to estimate the error of solution is proposed and the convergence of the method is proved. Finally, the validity and applicability of the proposed method is illustrated by some examples.
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    A comparative study on solving fractional cubic isothermal auto-catalytic chemical system via new efficient technique
    Bildik, N; Deniz, S; Saad, KM
    In this paper, we examine a cubic isothermal auto-catalytic chemical system (CIACS) with the help of the newly developed technique. Classical model of this system is transformed into a new fractional forms by using three different and special fractional operators. The new model is therefore called as fractional cubic isothermal auto-catalytic chemical system (FCIACS). Then, the new systems are solved by optimal perturbation iteration method. Obtained results are compared to get an idea about the new derivative operators and optimal perturbation iteration method. (C) 2019 Elsevier Ltd. All rights reserved.
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    New Iteration Methods for Time-Fractional Modified Nonlinear Kawahara Equation
    Atangana, A; Bildik, N; Noutchie, SCO
    We put side by side the methodology of two comparatively new analytical techniques to get to the bottom of the system of nonlinear fractional modified Kawahara equation. The technique is described and exemplified with a numerical example. The dependability of both methods and the lessening in computations give these methods a wider applicability. In addition, the computations implicated are very simple and undemanding.
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    On the numerical solution of initial value problems for nonlinear trapezoidal formulas with different types
    Bildik, N; Inç, M
    In this paper, the local truncation errors of the trapezoidal formulas such as arithmetic mean (AM), geometric mean (GM), heronian mean (HeM), harmonic mean (HaM), contraharmonic mean (CoM), root mean square (RMS), logarithmic mean (LM) and centroidal mean (CeM) are investigated and the stability analysis of these formulas are found. Finally, it is applied to various initial value problems.
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    Modified decomposition method for nonlinear Volterra-Fredholm integral equations
    Bildik, N; Inc, M
    In this paper, the nonlinear Volterra-Fredholm integral equations are solved by using the modified decomposition method (MDM). The approximate solution of this equation is calculated in the form of a series with easily computable components. The accuracy of the proposed numerical scheme is examined by comparison with other analytical and numerical results. Two test problems are presented to illustrate the reliability and the performance of the modified decomposition method. (c) 2006 Elsevier Ltd. All rights reserved.
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    On Common Coupled Fixed Point Theorems for Comparable Mappings in Ordered Partially Metric Spaces
    Mutlu, A; Yolcu, N; Mutlu, B; Bildik, N
    Common coupled fixed point theorems are examined in this paper for comparable mappings ensuring nonlinear contraction in ordered partial metric spaces. Given theorems enlarge and universalize some conclusions of Gnana Bhaskar and Lakshmikantham (2006).
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    New approximate solutions to electrostatic differential equations obtained by using numerical and analytical methods
    Bildik, N; Deniz, S
    In this paper, we implement the optimal homotopy asymptotic method to find the approximate solutions of the Poisson-Boltzmann equation. We also use the results of the conjugate gradient method for comparison with those of the optimal homotopy asymptotic method. Our study reveals that the optimal homotopy asymptotic method gives more effective results than conjugate gradient algorithms for the considered problems.
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    The solution of two dimensional nonlinear differential equation by the Adomian decomposition method
    Bildik, N; Bayramoglu, H
    In this paper the Adomian's decomposition method is used to investigate nonlinear two dimensional wave equation. The analytic Solution of the nonlinear wave equation is calculated in the form of a series with easily computable components. The nonhomogenous equation is effectively solved by employing the phenomena of the self-canceling ''noise terms where sum of components vanishes in the limit. Comparing the rnethodology with some known techniques shows that the present approach is powerful and reliable. Its remarkable accuracy properties are finally demonstrated by all example. (c) 2004 Elsevier Inc. All rights reserved.
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    A Practical Method for Analytical Evaluation of Approximate Solutions of Fisher's Equations
    Bildik, N; Deniz, S
    In this article, a framework is developed to get more approximate solutions to nonlinear partial differential equations by applying perturbation iteration technique. This technique is modified and improved to solve nonlinear diffusion equations of the Fisher type. Some problems are investigated to illustrate the efficiency of the method. Comparisons between the new results and the solutions obtained by other techniques prove that this technique is highly effective and accurate in solving nonlinear problems. Convergence analysis and error estimate are also provided by using some related theorems. The basic ideas indicated in this work are anticipated to be further developed to handle nonlinear models.
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    NEW APPROXIMATE SOLUTIONS TO THE NONLINEAR KLEIN-GORDON EQUATIONS USING PERTURBATION ITERATION TECHNIQUES
    Bildik, N; Deniz, S
    In this study, we present the new approximate solutions of the nonlinear Klein-Gordon equations via perturbation iteration technique and newly developed optimal perturbation iteration method. Some specific examples are given and obtained solutions are compared with other methods and analytical results to confirm the good accuracy of the proposed methods.We also discuss the convergence of the optimal perturbation iteration method for partial differential equations. The results reveal that perturbation iteration techniques,unlike many other techniques in literature, converge rapidly to exact solutions of the given problems at lower order of approximations.
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    A new analytical technique for solving Lane - Emden type equations arising in astrophysics
    Deniz, S; Bildik, N
    Lane - Emden type equations are nonlinear differential equations which represent many scientific phenomena in astrophysics and mathematical physics. In this study, a new analytic approximate technique for addressing nonlinear problems, namely the optimal perturbation iteration method, is introduced and implemented to singular initial value Lane-Emden type problems to test the effectiveness and performance of the method. This technique provides us to adjust the convergence regions when necessary. Comparing different methods reveals that the proposed method is highly accurate and has great potential to be a new kind of powerful analytical tool for Lane Emden type equations.
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    A new fractional analysis on the polluted lakes system
    Bildik, N; Deniz, S
    In this paper, we use Atangana-Baleanu derivative which is defined with the Mittag-Leffler function and has all the properties of a classical fractional derivative for solving the system of fractional differential equations. The classical model of polluted lakes system is modified by using the concept of fractional differentiation with nonsingular and nonlocal fading memory. The new numerical scheme recommended by Toufik and Atangana is used to analyze the modified model of polluted lakes system. Some numerical illustrations are presented to show the effect of the new fractional differentiation. (C) 2019 Elsevier Ltd. All rights reserved.