Necdet BİLDİK - Profile on Academia.edu (original) (raw)

Papers by Necdet BİLDİK

Scientia Iranica

In this paper, we used the Picard successive iteration method and the new modified Krasnoselskii ... more In this paper, we used the Picard successive iteration method and the new modified Krasnoselskii iteration method in order to solve different types of ordinary linear differential equations having initial conditions. By applying the new modified Krasnoselskii iteration method, not only do we obtain the approximate solutions for the problem, but also establish the corresponding iterative schemes. Finally, it is shown that the accuracy of the new iteration method (called the new modified Krasnoselskii iteration method) is substantially improved by employing variable steps which adjust themselves to the solution of the differential equation.

International Journal of Applied Physics and Mathematics, 2017

In this study, we examine the approximate solutions of complex differential equations in rectangu... more In this study, we examine the approximate solutions of complex differential equations in rectangular domains by using Euler polynomials. We construct the matrix forms of Euler polynomials and their derivatives to transform the considered differential equation to matrix equation with unknown Euler coefficients. This matrix equation is also equivalent to a system of linear algebraic equations. Linear system is solved by substituting collocation points into those matrix forms to get the unknown Euler coefficients. Determining these coefficients provides the approximate solutions of the given complex differential equations under the given conditions.

Celal Bayar Universitesi Fen Bilimleri Dergisi, 2017

In this paper, the first-order ODEs which have no systematic way to find their Lie point symmetri... more In this paper, the first-order ODEs which have no systematic way to find their Lie point symmetries -unlike higher order ODEs which have systematic ways- are reconsidered. As a first step, we considered first order PDEs which correspond to these equations by introducing reduced characteristic Q that used in the Lie’s theory. Following this step, we tried to obtain solutions of the PDEs using their Lie point symmetries. But in this process, we met some difficulties, so by taking into account some assumptions we obtained the symmetries of ODEs which are in the special form, and also their solutions.

In this paper, solution of systems of delay di erential equations, with initial conditions, using... more In this paper, solution of systems of delay di erential equations, with initial conditions, using numerical methods, including the Taylor collocation method, the Lambert W function and the variational iteration method, is considered. We have endeavored to show the most appropriate method by comparing the solutions of this system of equations with di erent types of methods. All numerical computations have been performed on the computer algebraic system, Matlab.

arXiv: Classical Analysis and ODEs, 2016

A new analytic approximate technique for addressing nonlinear problems, namely the optimal pertur... more 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 nonlinear differential equations.

Iranian Journal of Science and Technology-Transactions of Mechanical Engineering, 2015

In this paper, the dynamical behavior of an axially moving string modeled by fractional derivativ... more In this paper, the dynamical behavior of an axially moving string modeled by fractional derivative is investigated. The governing equation represented motion is solved by the method of multiple scales. Considering principal parametric resonance, the stability boundaries for string with simple supports are obtained. Numerical results indicate the effects of fractional damping on stability.

In this study, we solve Riccati differential equations by modified Adomian decomposition method w... more In this study, we solve Riccati differential equations by modified Adomian decomposition method which is constructed by different orthogonal polynomials. Here, Chebyshev polynomials are used instead of Taylor polynomials to expand the source function. We see the benefits of using these expansions to get better results.

In this research paper, a different semi-analytical analysis of modified magnetohydrodynamic Jeff... more In this research paper, a different semi-analytical analysis of modified magnetohydrodynamic Jeffery–Hamel flow is conducted via the newly developed technique. We use the optimal iterative perturbation method with multiple parameters to see the effects of the magnetic field and nanoparticle on the Jeffery–Hamel flow. Comparing our new approximate solutions with some earlier works proved the excellent accuracy of the newly proposed technique. Convergence analysis of the proposed method is also discussed and error estimation is given to anticipate the accuracy of higher-order approximate solutions.

Advances in Mathematics: Scientific Journal

In this study, we recover potential function and separable boundary conditions for the inverse St... more In this study, we recover potential function and separable boundary conditions for the inverse Sturm-Liouville problem in normal form by using two partial subsets of the data which consist of its one spectrum and sequence of endpoints of eigenfunctions.

Legendre wavelet solution of neutral differential equations with proportional delays

Journal of Applied Mathematics and Computing

Discrete & Continuous Dynamical Systems - S

In this study, we present the new approximate solutions of the nonlinear Klein-Gordon equations v... more 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.

Georgian Mathematical Journal

In this paper, we implement the optimal homotopy asymptotic method to find the approximate soluti... more 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.

ITM Web of Conferences

In this article, a framework is developed to get more approximate solutions to nonlinear partial ... more 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.

Solving the burgers' and regularized long wave equations using the new perturbation iteration technique

Numerical Methods for Partial Differential Equations

International Journal of Applied Physics and Mathematics

In this study, optimal perturbation iteration method is implemented to solve Korteweg de Vries (K... more In this study, optimal perturbation iteration method is implemented to solve Korteweg de Vries (KdV)-like equation to obtain semi analytical solutions. We examine two illustrations to analyze the new optimal perturbation iteration method. This work displays that optimal perturbation iteration technique converges fast to the exact solutions of the differential equations at lower order of approximations.

Journal of Fuzzy Set Valued Analysis

In this study, we examine fuzzy differential equations by using a new and dynamic approach. A nov... more In this study, we examine fuzzy differential equations by using a new and dynamic approach. A novel scheme based on logarithmic mean is discussed in detail and comparison with harmonic mean is also performed with error analysis. Obtained solutions reveal that one gets very accurate and effective results by applying this scheme to solve the fuzzy differential equations.

A new efficient method for solving delay differential equations and a comparison with other methods

The European Physical Journal Plus

International Journal of Applied Physics and Mathematics

In this work, we obtain the approximate solutions for the telegraph equations by using optimal pe... more In this work, we obtain the approximate solutions for the telegraph equations by using optimal perturbation iteration technique. We consider two examples to illustrate the proposed method. The present paper also unveils that optimal perturbation iteration technique converges fast to the accurate solutions of the given equations at lower order of iterations.

International Journal of Modeling and Optimization

In this study, the concept of Homotopy analysis method (HAM) is briefly introduced. Furthermore s... more In this study, the concept of Homotopy analysis method (HAM) is briefly introduced. Furthermore some non-linear problems are handled and the solutions of these problems are given using by HAM, DTM, ADM methods and the convergence of the solution is shown to the exact solution. Additionally, the three methods are compared and it is observed that the HAM is more-less efficient and effective than the ADM and DTM in according to the exact solution. In the end, some of the numerical solutions of two examples are presented and the results are shown in graphs and figures.

Celal Bayar Üniversitesi Fen Bilimleri Dergisi, 2017

In this study, we investigate the stability of Taylor collocation method for initial value proble... more In this study, we investigate the stability of Taylor collocation method for initial value problems in ordinary differential equations. Firstly, we try to show that Taylor collocation method for initial value problem is equivalent to a subset of the implicit Runge-Kutta methods. This equivalence enables us to prove that Taylor collocation method is absolutely stable (A-stable) for the considered equations.

Scientia Iranica

In this paper, we used the Picard successive iteration method and the new modified Krasnoselskii ... more In this paper, we used the Picard successive iteration method and the new modified Krasnoselskii iteration method in order to solve different types of ordinary linear differential equations having initial conditions. By applying the new modified Krasnoselskii iteration method, not only do we obtain the approximate solutions for the problem, but also establish the corresponding iterative schemes. Finally, it is shown that the accuracy of the new iteration method (called the new modified Krasnoselskii iteration method) is substantially improved by employing variable steps which adjust themselves to the solution of the differential equation.

International Journal of Applied Physics and Mathematics, 2017

In this study, we examine the approximate solutions of complex differential equations in rectangu... more In this study, we examine the approximate solutions of complex differential equations in rectangular domains by using Euler polynomials. We construct the matrix forms of Euler polynomials and their derivatives to transform the considered differential equation to matrix equation with unknown Euler coefficients. This matrix equation is also equivalent to a system of linear algebraic equations. Linear system is solved by substituting collocation points into those matrix forms to get the unknown Euler coefficients. Determining these coefficients provides the approximate solutions of the given complex differential equations under the given conditions.

Celal Bayar Universitesi Fen Bilimleri Dergisi, 2017

In this paper, the first-order ODEs which have no systematic way to find their Lie point symmetri... more In this paper, the first-order ODEs which have no systematic way to find their Lie point symmetries -unlike higher order ODEs which have systematic ways- are reconsidered. As a first step, we considered first order PDEs which correspond to these equations by introducing reduced characteristic Q that used in the Lie’s theory. Following this step, we tried to obtain solutions of the PDEs using their Lie point symmetries. But in this process, we met some difficulties, so by taking into account some assumptions we obtained the symmetries of ODEs which are in the special form, and also their solutions.

In this paper, solution of systems of delay di erential equations, with initial conditions, using... more In this paper, solution of systems of delay di erential equations, with initial conditions, using numerical methods, including the Taylor collocation method, the Lambert W function and the variational iteration method, is considered. We have endeavored to show the most appropriate method by comparing the solutions of this system of equations with di erent types of methods. All numerical computations have been performed on the computer algebraic system, Matlab.

arXiv: Classical Analysis and ODEs, 2016

A new analytic approximate technique for addressing nonlinear problems, namely the optimal pertur... more 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 nonlinear differential equations.

Iranian Journal of Science and Technology-Transactions of Mechanical Engineering, 2015

In this paper, the dynamical behavior of an axially moving string modeled by fractional derivativ... more In this paper, the dynamical behavior of an axially moving string modeled by fractional derivative is investigated. The governing equation represented motion is solved by the method of multiple scales. Considering principal parametric resonance, the stability boundaries for string with simple supports are obtained. Numerical results indicate the effects of fractional damping on stability.

In this study, we solve Riccati differential equations by modified Adomian decomposition method w... more In this study, we solve Riccati differential equations by modified Adomian decomposition method which is constructed by different orthogonal polynomials. Here, Chebyshev polynomials are used instead of Taylor polynomials to expand the source function. We see the benefits of using these expansions to get better results.

In this research paper, a different semi-analytical analysis of modified magnetohydrodynamic Jeff... more In this research paper, a different semi-analytical analysis of modified magnetohydrodynamic Jeffery–Hamel flow is conducted via the newly developed technique. We use the optimal iterative perturbation method with multiple parameters to see the effects of the magnetic field and nanoparticle on the Jeffery–Hamel flow. Comparing our new approximate solutions with some earlier works proved the excellent accuracy of the newly proposed technique. Convergence analysis of the proposed method is also discussed and error estimation is given to anticipate the accuracy of higher-order approximate solutions.

Advances in Mathematics: Scientific Journal

In this study, we recover potential function and separable boundary conditions for the inverse St... more In this study, we recover potential function and separable boundary conditions for the inverse Sturm-Liouville problem in normal form by using two partial subsets of the data which consist of its one spectrum and sequence of endpoints of eigenfunctions.

Legendre wavelet solution of neutral differential equations with proportional delays

Journal of Applied Mathematics and Computing

Discrete & Continuous Dynamical Systems - S

In this study, we present the new approximate solutions of the nonlinear Klein-Gordon equations v... more 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.

Georgian Mathematical Journal

In this paper, we implement the optimal homotopy asymptotic method to find the approximate soluti... more 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.

ITM Web of Conferences

In this article, a framework is developed to get more approximate solutions to nonlinear partial ... more 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.

Solving the burgers' and regularized long wave equations using the new perturbation iteration technique

Numerical Methods for Partial Differential Equations

International Journal of Applied Physics and Mathematics

In this study, optimal perturbation iteration method is implemented to solve Korteweg de Vries (K... more In this study, optimal perturbation iteration method is implemented to solve Korteweg de Vries (KdV)-like equation to obtain semi analytical solutions. We examine two illustrations to analyze the new optimal perturbation iteration method. This work displays that optimal perturbation iteration technique converges fast to the exact solutions of the differential equations at lower order of approximations.

Journal of Fuzzy Set Valued Analysis

In this study, we examine fuzzy differential equations by using a new and dynamic approach. A nov... more In this study, we examine fuzzy differential equations by using a new and dynamic approach. A novel scheme based on logarithmic mean is discussed in detail and comparison with harmonic mean is also performed with error analysis. Obtained solutions reveal that one gets very accurate and effective results by applying this scheme to solve the fuzzy differential equations.

A new efficient method for solving delay differential equations and a comparison with other methods

The European Physical Journal Plus

International Journal of Applied Physics and Mathematics

In this work, we obtain the approximate solutions for the telegraph equations by using optimal pe... more In this work, we obtain the approximate solutions for the telegraph equations by using optimal perturbation iteration technique. We consider two examples to illustrate the proposed method. The present paper also unveils that optimal perturbation iteration technique converges fast to the accurate solutions of the given equations at lower order of iterations.

International Journal of Modeling and Optimization

In this study, the concept of Homotopy analysis method (HAM) is briefly introduced. Furthermore s... more In this study, the concept of Homotopy analysis method (HAM) is briefly introduced. Furthermore some non-linear problems are handled and the solutions of these problems are given using by HAM, DTM, ADM methods and the convergence of the solution is shown to the exact solution. Additionally, the three methods are compared and it is observed that the HAM is more-less efficient and effective than the ADM and DTM in according to the exact solution. In the end, some of the numerical solutions of two examples are presented and the results are shown in graphs and figures.

Celal Bayar Üniversitesi Fen Bilimleri Dergisi, 2017

In this study, we investigate the stability of Taylor collocation method for initial value proble... more In this study, we investigate the stability of Taylor collocation method for initial value problems in ordinary differential equations. Firstly, we try to show that Taylor collocation method for initial value problem is equivalent to a subset of the implicit Runge-Kutta methods. This equivalence enables us to prove that Taylor collocation method is absolutely stable (A-stable) for the considered equations.