Şuayip Yüzbaşı | Akdeniz University (original) (raw)

Papers by Şuayip Yüzbaşı

Research paper thumbnail of An operational matrix method for solving linear Fredholm--Volterra integro-differential equations

TURKISH JOURNAL OF MATHEMATICS, 2018

The aim of this paper is to propose an efficient method to compute approximate solutions of linea... more The aim of this paper is to propose an efficient method to compute approximate solutions of linear Fredholm-Volterra integro-differential equations (FVIDEs) using Taylor polynomials. More precisely, we present a method based on operational matrices of Taylor polynomials in order to solve linear FVIDEs. By using the operational matrices of integration and product for the Taylor polynomials, the problem for linear FVIDEs is transformed into a system of linear algebraic equations. The solution of the problem is obtained from this linear system after the incorporation of initial conditions. Numerical examples are presented to show the applicability and the efficiency of the method. Wherever possible, the results of our method are compared with those yielded by some other methods.

Research paper thumbnail of A new Bell function approach to solve linear fractional differential equations

Applied Numerical Mathematics

Research paper thumbnail of A Laguerre Approach for Solving of the Systems of Linear Differential Equations and Residual Improvement

Computational Methods for Differential Equations, 2020

In this study, a collocation method based on Laguerre polynomials is presented to numerically sol... more In this study, a collocation method based on Laguerre polynomials is presented to numerically solve systems of linear differential equations with variable coefficients of high order. The method contains the following steps. Firstly, we write the Laguerre polynomials, their derivatives and the solutions in matrix form. Secondly, the system of linear differential equations is reduced to a system of linear algebraic equations by means of matrix relations and collocation points. Then, the conditions in the problem are also written in the form of matrix of Laguerre polynomials. Hence, by using the obtained algebraic system and the matrix form of the conditions, a new system of linear algebraic equation is obtained. By solving the system of the obtained new algebraic equation, the coef- cients of the approximate solution of the problem are determined. For the problem, the residual error estimation technique is offered and approximate solutions are improved. Finally, the presented method a...

Research paper thumbnail of Application of Vieta–Lucas Series to Solve a Class of Multi-Pantograph Delay Differential Equations with Singularity

Symmetry

The main focus of this paper was to find the approximate solution of a class of second-order mult... more The main focus of this paper was to find the approximate solution of a class of second-order multi-pantograph delay differential equations with singularity. We used the shifted version of Vieta–Lucas polynomials with some symmetries as the main base to develop a collocation approach for solving the aforementioned differential equations. Moreover, an error bound of the present approach by using the maximum norm was computed and an error estimation technique based on the residual function is presented. Finally, the validity and applicability of the presented collocation scheme are shown via four numerical test examples.

Research paper thumbnail of Approximate solutions of the hyperchaotic Rössler system by using the Bessel collocation scheme

The purpose of this study is to give a Bessel polynomial approximation for the solutions of the h... more The purpose of this study is to give a Bessel polynomial approximation for the solutions of the hyperchaotic R¨ossler system.For this purpose, the Bessel collocation method applied to different problems is developed for the mentioned system. This method isbased on taking the truncated Bessel expansions of the functions in the hyperchaotic R¨ossler systems. The suggested secheme convertsthe problem into a system of nonlinear algebraic equations by means of the matrix operations and collocation points, The accuracy andefficiency of the proposed approach are demonstrated by numerical applications and performed with the help of a computer code writtenin Maple. Also, comparison between our method and the differential transformation method is made with the accuracy of solutions

Research paper thumbnail of Laguerre Collocation Method for Solutions of Systems of First Order Linear Differential Equations

In this paper, a collocation method based on Laguerre polynomials is presented to solve systems o... more In this paper, a collocation method based on Laguerre polynomials is presented to solve systems of linear differential equations. The Laguerre polynomials, their derivatives, system of differential equations and conditions are written in the matrix form. Then, by using the constructed matrix forms, collocation points and matrix operations, the system of linear differential equations is transformed into a system of linear algebraic equations. The solution of this system gives the coefficients of the solutions forms. Thus, the solutions based on the Laguerre polynomials is found. Also, error estimation is made by using residual functions. Numerical examples are given to explain the method. The results are compared with results of other methods.

Research paper thumbnail of An operational matrix method to solve the Lotka–Volterra predator–prey models with discrete delays

Chaos, Solitons & Fractals

Research paper thumbnail of A Computational Algorithm for Solution of Population Models for Single and Interacting Species

International Journal of Applied and Computational Mathematics

In this paper, we present a computational technique based on Haar wavelet for two continuous popu... more In this paper, we present a computational technique based on Haar wavelet for two continuous population models (CPMs) regarding single and interacting species. The derivative involved in the population model is approximated using Haar functions in the Haar collocation technique, and the integration process is used to obtain the estimated solution for the unknown function involved in a population model. Also the error estimation of the proposed technique for CPMs for single and interacting species are given to check the accuracy of proposed technique. To demonstrate the accuracy of the proposed technique for single and interacting species, some examples are given. The rate of convergence is also estimated, which is approximately equal to 2, confirming the theoretical results. The results are compared with the exact solution, and the technique efficiency is demonstrated by measuring maximum absolute errors using different collocation points. The results show that Haar technique is simple and robust.

Research paper thumbnail of Approximating solutions to fractional-order Bagley-Torvik equation via generalized Bessel polynomial on large domains

Research paper thumbnail of Approximating Solutions of Non-Linear Troesch’s Problem via an Efficient Quasi-Linearization Bessel Approach

Mathematics

Two collocation-based methods utilizing the novel Bessel polynomials (with positive coefficients)... more Two collocation-based methods utilizing the novel Bessel polynomials (with positive coefficients) are developed for solving the non-linear Troesch’s problem. In the first approach, by expressing the unknown solution and its second derivative in terms of the Bessel matrix form along with some collocation points, the governing equation transforms into a non-linear algebraic matrix equation. In the second approach, the technique of quasi-linearization is first employed to linearize the model problem and, then, the first collocation method is applied to the sequence of linearized equations iteratively. In the latter approach, we require to solve a linear algebraic matrix equation in each iteration. Moreover, the error analysis of the Bessel series solution is established. In the end, numerical simulations and computational results are provided to illustrate the utility and applicability of the presented collocation approaches. Numerical comparisons with some existing available methods a...

Research paper thumbnail of A Taylor–Chebyshev approximation technique to solve the 1D and 2D nonlinear Burgers equations

Mathematical Sciences

This paper deals with proposing an approximate solution for the well-known Burgers equation as a ... more This paper deals with proposing an approximate solution for the well-known Burgers equation as a canonical model of various fields of science and engineering. Our novel combined approximation algorithm is based on the linearized Taylor approach for the time discretization, while the spectral Chebyshev collocation method is utilized for the space variables. This implies that in each time step, the proposed combined approach reduces the one- and two-dimensional model problems into a system of linear equations, which consists of polynomial coefficients. The error analysis of the present approach in 1D and 2D is discussed. Through numerical simulations, the utility and efficiency of the combined scheme are examined and comparisons with exact solutions as well as existing available methods have been performed. The comparisons indicate that the combined approach is efficient, practical, and straightforward in implementation. The technique developed can be easily extended to other nonlinear models.

Research paper thumbnail of On the solution of the Human Immunodeficiency Virus (HIV) infection model using spectral collocation method

International Journal of Biomathematics

In this research work, we study the Human Immunodeficiency Virus (HIV) infection on helper T cell... more In this research work, we study the Human Immunodeficiency Virus (HIV) infection on helper T cells governed by a mathematical model consisting of a system of three first-order nonlinear differential equations. The objective of the analysis is to present an approximate mathematical solution to the model that gives the count of the numbers of uninfected and infected helper T cells and the number of free virus particles present at a given instant of time. The system of nonlinear ODEs is converted into a system of nonlinear algebraic equations using spectral collocation method with three different basis functions such as Chebyshev, Legendre and Jacobi polynomials. Some factors such as the production of helper T cells and infection of these cells play a vital role in infected and uninfected cell counts. Detailed error analysis is done to compare our results with the existing methods. It is shown that the spectral collocation method is a very reliable, efficient and robust method of solut...

Research paper thumbnail of A Galerkin-Type Fractional Approach for Solutions of Bagley-Torvik Equations

Computer Modeling in Engineering & Sciences

Research paper thumbnail of Introduction to the Special Issue on Numerical Methods for Differential and Integral Equations

Computer Modeling in Engineering & Sciences

Research paper thumbnail of An operational method for solutions of Riccati type differential equations with functional arguments

Journal of Taibah University for Science

In this article, an operational matrix approach is presented to solve the Riccati type differenti... more In this article, an operational matrix approach is presented to solve the Riccati type differential equations with functional arguments. These equations are encountered in Mathematical Physics. The method is based on the least-squares approximation and the operational matrices of integration and product. By obtaining the operation matrices for each term of the problem, the method converts the problem to a system of nonlinear algebraic equations. The roots of last system are used in determination of unknown function. Error analysis is made. Numerical applications are given to show efficiency of the method and also the comparisons are made with other methods from literature. In applications of the method, it is observed from the applications that the suggested method gives effective results.

Research paper thumbnail of An exponential method to solve linear Fredholm–Volterra integro-differential equations and residual improvement

TURKISH JOURNAL OF MATHEMATICS

In this paper, a collocation approach based on exponential polynomials is introduced to solve lin... more In this paper, a collocation approach based on exponential polynomials is introduced to solve linear Fredholm-Volterra integro-differential equations under the initial boundary conditions. First, by constructing the matrix forms of the exponential polynomials and their derivatives, the desired exponential solution and its derivatives are written in matrix forms. Second, the differential and integral parts of the problem are converted into matrix forms based on exponential polynomials. Later, the main problem is reduced to a system of linear algebraic equations by aid of the collocation points, the matrix operations, and the matrix forms of the conditions. The solutions of this system give the coefficients of the desired exponential solution. An error estimation method is also presented by using the residual function and the exponential solutions are improved by the estimated error function. Numerical examples are solved to show the applicability and the effectiveness of the method. In addition, the results are compared with the results of other methods.

Research paper thumbnail of Pell–Lucas collocation method for numerical solutions of two population models and residual correction

Journal of Taibah University for Science

Our aim in this article is to present a collocation method to solve two population models for sin... more Our aim in this article is to present a collocation method to solve two population models for single and interacting species. For this, logistic growth model and prey-predator model are examined. These models are solved numerically by Pell-Lucas collocation method. The method gives the approximate solutions of these models in form of truncated Pell-Lucas series. By utilizing Pell-Lucas collocation method, non-linear mathematical models are converted to a system of non-linear algebraic equations. This non-linear equation system is solved and the obtained coefficients are the coefficients of the truncated Pell-Lucas serie solution. Furthermore, the residual correction method is used to find better approximate solutions. All results are shown in tables and graphs for different (N, M) values, and additionally the comparisons are made with other methods from. It is seen that the method gives effective results to the presented model problems.

Research paper thumbnail of Bessel collocation approach for approximate solutions of Hantavirus infection model

New Trends in Mathematical Science, Nov 8, 2017

In this study, a collocation method is introduced to find the approximate solutions of Hantavirus... more In this study, a collocation method is introduced to find the approximate solutions of Hantavirus infection model which is a system of nonlinear ordinary differential equations. The method is based on the Bessel functions of the first kind, matrix operations and collocation points. This method converts Hantavirus infection model into a matrix equation in terms of the Bessel functions of first kind, matrix operations and collocation points. The matrix equation corresponds to a system of nonlinear equations with the unknown Bessel coefficients. The reliability and efficiency of the suggested scheme are demonstrated by numerical applications and all numerical calculations have been done by using a program written in Maple.

Research paper thumbnail of Algorithm for the Numerical Solutions of Volterra Population Growth Model with Fractional Order via Haar Wavelet

Contemporary Mathematics

In this paper, a new collocation method based on Haar wavelet is developed for the numerical solu... more In this paper, a new collocation method based on Haar wavelet is developed for the numerical solution ofthe fractional Volterra model (FVM) for population growth of a species in a closed system. In the proposed method the derivative involved in the nonlinear model is approximated using Haar wavelet and the approximate expressions for the unknown function is obtained by the process of integration, the fractional derivative will be considered in the Caputo sense. The technique of residual correction, which aims to reduce the error of the approximate solution by estimating this error, is discussed in some detail. To show the computational efficiency of the proposed method, the residual correction technique are illustrated with an example. The numerical results are compared with existing methods from the literature. The numerical results show that the method is simply applicable, accurate, efficient and robust.

Research paper thumbnail of Pell-Lucas collocation method to solve high-order linear Fredholm-Volterra integro-differential equations and residual correction

TURKISH JOURNAL OF MATHEMATICS

In this article, a collocation method based on Pell-Lucas polynomials is studied to numerically s... more In this article, a collocation method based on Pell-Lucas polynomials is studied to numerically solve higher order linear Fredholm-Volterra integro differential equations (FVIDE). The approximate solutions are assumed in form of the truncated Pell-Lucas polynomial series. By using Pell-Lucas polynomials and relations of their derivatives, the solution form and its derivatives are brought to matrix forms. By applying the collocation method based on equally spaced collocation points, the method reduces the problem to a system of linear algebraic equations. Solution of this system determines the coefficients of assumed solution. Error estimation is made and also a method with the help of the obtained approximate solution is developed that finds approximate solution with better results. Then, the applications are made on five examples to show that the method is successful. In addition, the results are supported by tables and graphs and the comparisons are made with other methods available in the literature. All calculations in this study have been made using codes written in Matlab.

Research paper thumbnail of An operational matrix method for solving linear Fredholm--Volterra integro-differential equations

TURKISH JOURNAL OF MATHEMATICS, 2018

The aim of this paper is to propose an efficient method to compute approximate solutions of linea... more The aim of this paper is to propose an efficient method to compute approximate solutions of linear Fredholm-Volterra integro-differential equations (FVIDEs) using Taylor polynomials. More precisely, we present a method based on operational matrices of Taylor polynomials in order to solve linear FVIDEs. By using the operational matrices of integration and product for the Taylor polynomials, the problem for linear FVIDEs is transformed into a system of linear algebraic equations. The solution of the problem is obtained from this linear system after the incorporation of initial conditions. Numerical examples are presented to show the applicability and the efficiency of the method. Wherever possible, the results of our method are compared with those yielded by some other methods.

Research paper thumbnail of A new Bell function approach to solve linear fractional differential equations

Applied Numerical Mathematics

Research paper thumbnail of A Laguerre Approach for Solving of the Systems of Linear Differential Equations and Residual Improvement

Computational Methods for Differential Equations, 2020

In this study, a collocation method based on Laguerre polynomials is presented to numerically sol... more In this study, a collocation method based on Laguerre polynomials is presented to numerically solve systems of linear differential equations with variable coefficients of high order. The method contains the following steps. Firstly, we write the Laguerre polynomials, their derivatives and the solutions in matrix form. Secondly, the system of linear differential equations is reduced to a system of linear algebraic equations by means of matrix relations and collocation points. Then, the conditions in the problem are also written in the form of matrix of Laguerre polynomials. Hence, by using the obtained algebraic system and the matrix form of the conditions, a new system of linear algebraic equation is obtained. By solving the system of the obtained new algebraic equation, the coef- cients of the approximate solution of the problem are determined. For the problem, the residual error estimation technique is offered and approximate solutions are improved. Finally, the presented method a...

Research paper thumbnail of Application of Vieta–Lucas Series to Solve a Class of Multi-Pantograph Delay Differential Equations with Singularity

Symmetry

The main focus of this paper was to find the approximate solution of a class of second-order mult... more The main focus of this paper was to find the approximate solution of a class of second-order multi-pantograph delay differential equations with singularity. We used the shifted version of Vieta–Lucas polynomials with some symmetries as the main base to develop a collocation approach for solving the aforementioned differential equations. Moreover, an error bound of the present approach by using the maximum norm was computed and an error estimation technique based on the residual function is presented. Finally, the validity and applicability of the presented collocation scheme are shown via four numerical test examples.

Research paper thumbnail of Approximate solutions of the hyperchaotic Rössler system by using the Bessel collocation scheme

The purpose of this study is to give a Bessel polynomial approximation for the solutions of the h... more The purpose of this study is to give a Bessel polynomial approximation for the solutions of the hyperchaotic R¨ossler system.For this purpose, the Bessel collocation method applied to different problems is developed for the mentioned system. This method isbased on taking the truncated Bessel expansions of the functions in the hyperchaotic R¨ossler systems. The suggested secheme convertsthe problem into a system of nonlinear algebraic equations by means of the matrix operations and collocation points, The accuracy andefficiency of the proposed approach are demonstrated by numerical applications and performed with the help of a computer code writtenin Maple. Also, comparison between our method and the differential transformation method is made with the accuracy of solutions

Research paper thumbnail of Laguerre Collocation Method for Solutions of Systems of First Order Linear Differential Equations

In this paper, a collocation method based on Laguerre polynomials is presented to solve systems o... more In this paper, a collocation method based on Laguerre polynomials is presented to solve systems of linear differential equations. The Laguerre polynomials, their derivatives, system of differential equations and conditions are written in the matrix form. Then, by using the constructed matrix forms, collocation points and matrix operations, the system of linear differential equations is transformed into a system of linear algebraic equations. The solution of this system gives the coefficients of the solutions forms. Thus, the solutions based on the Laguerre polynomials is found. Also, error estimation is made by using residual functions. Numerical examples are given to explain the method. The results are compared with results of other methods.

Research paper thumbnail of An operational matrix method to solve the Lotka–Volterra predator–prey models with discrete delays

Chaos, Solitons & Fractals

Research paper thumbnail of A Computational Algorithm for Solution of Population Models for Single and Interacting Species

International Journal of Applied and Computational Mathematics

In this paper, we present a computational technique based on Haar wavelet for two continuous popu... more In this paper, we present a computational technique based on Haar wavelet for two continuous population models (CPMs) regarding single and interacting species. The derivative involved in the population model is approximated using Haar functions in the Haar collocation technique, and the integration process is used to obtain the estimated solution for the unknown function involved in a population model. Also the error estimation of the proposed technique for CPMs for single and interacting species are given to check the accuracy of proposed technique. To demonstrate the accuracy of the proposed technique for single and interacting species, some examples are given. The rate of convergence is also estimated, which is approximately equal to 2, confirming the theoretical results. The results are compared with the exact solution, and the technique efficiency is demonstrated by measuring maximum absolute errors using different collocation points. The results show that Haar technique is simple and robust.

Research paper thumbnail of Approximating solutions to fractional-order Bagley-Torvik equation via generalized Bessel polynomial on large domains

Research paper thumbnail of Approximating Solutions of Non-Linear Troesch’s Problem via an Efficient Quasi-Linearization Bessel Approach

Mathematics

Two collocation-based methods utilizing the novel Bessel polynomials (with positive coefficients)... more Two collocation-based methods utilizing the novel Bessel polynomials (with positive coefficients) are developed for solving the non-linear Troesch’s problem. In the first approach, by expressing the unknown solution and its second derivative in terms of the Bessel matrix form along with some collocation points, the governing equation transforms into a non-linear algebraic matrix equation. In the second approach, the technique of quasi-linearization is first employed to linearize the model problem and, then, the first collocation method is applied to the sequence of linearized equations iteratively. In the latter approach, we require to solve a linear algebraic matrix equation in each iteration. Moreover, the error analysis of the Bessel series solution is established. In the end, numerical simulations and computational results are provided to illustrate the utility and applicability of the presented collocation approaches. Numerical comparisons with some existing available methods a...

Research paper thumbnail of A Taylor–Chebyshev approximation technique to solve the 1D and 2D nonlinear Burgers equations

Mathematical Sciences

This paper deals with proposing an approximate solution for the well-known Burgers equation as a ... more This paper deals with proposing an approximate solution for the well-known Burgers equation as a canonical model of various fields of science and engineering. Our novel combined approximation algorithm is based on the linearized Taylor approach for the time discretization, while the spectral Chebyshev collocation method is utilized for the space variables. This implies that in each time step, the proposed combined approach reduces the one- and two-dimensional model problems into a system of linear equations, which consists of polynomial coefficients. The error analysis of the present approach in 1D and 2D is discussed. Through numerical simulations, the utility and efficiency of the combined scheme are examined and comparisons with exact solutions as well as existing available methods have been performed. The comparisons indicate that the combined approach is efficient, practical, and straightforward in implementation. The technique developed can be easily extended to other nonlinear models.

Research paper thumbnail of On the solution of the Human Immunodeficiency Virus (HIV) infection model using spectral collocation method

International Journal of Biomathematics

In this research work, we study the Human Immunodeficiency Virus (HIV) infection on helper T cell... more In this research work, we study the Human Immunodeficiency Virus (HIV) infection on helper T cells governed by a mathematical model consisting of a system of three first-order nonlinear differential equations. The objective of the analysis is to present an approximate mathematical solution to the model that gives the count of the numbers of uninfected and infected helper T cells and the number of free virus particles present at a given instant of time. The system of nonlinear ODEs is converted into a system of nonlinear algebraic equations using spectral collocation method with three different basis functions such as Chebyshev, Legendre and Jacobi polynomials. Some factors such as the production of helper T cells and infection of these cells play a vital role in infected and uninfected cell counts. Detailed error analysis is done to compare our results with the existing methods. It is shown that the spectral collocation method is a very reliable, efficient and robust method of solut...

Research paper thumbnail of A Galerkin-Type Fractional Approach for Solutions of Bagley-Torvik Equations

Computer Modeling in Engineering & Sciences

Research paper thumbnail of Introduction to the Special Issue on Numerical Methods for Differential and Integral Equations

Computer Modeling in Engineering & Sciences

Research paper thumbnail of An operational method for solutions of Riccati type differential equations with functional arguments

Journal of Taibah University for Science

In this article, an operational matrix approach is presented to solve the Riccati type differenti... more In this article, an operational matrix approach is presented to solve the Riccati type differential equations with functional arguments. These equations are encountered in Mathematical Physics. The method is based on the least-squares approximation and the operational matrices of integration and product. By obtaining the operation matrices for each term of the problem, the method converts the problem to a system of nonlinear algebraic equations. The roots of last system are used in determination of unknown function. Error analysis is made. Numerical applications are given to show efficiency of the method and also the comparisons are made with other methods from literature. In applications of the method, it is observed from the applications that the suggested method gives effective results.

Research paper thumbnail of An exponential method to solve linear Fredholm–Volterra integro-differential equations and residual improvement

TURKISH JOURNAL OF MATHEMATICS

In this paper, a collocation approach based on exponential polynomials is introduced to solve lin... more In this paper, a collocation approach based on exponential polynomials is introduced to solve linear Fredholm-Volterra integro-differential equations under the initial boundary conditions. First, by constructing the matrix forms of the exponential polynomials and their derivatives, the desired exponential solution and its derivatives are written in matrix forms. Second, the differential and integral parts of the problem are converted into matrix forms based on exponential polynomials. Later, the main problem is reduced to a system of linear algebraic equations by aid of the collocation points, the matrix operations, and the matrix forms of the conditions. The solutions of this system give the coefficients of the desired exponential solution. An error estimation method is also presented by using the residual function and the exponential solutions are improved by the estimated error function. Numerical examples are solved to show the applicability and the effectiveness of the method. In addition, the results are compared with the results of other methods.

Research paper thumbnail of Pell–Lucas collocation method for numerical solutions of two population models and residual correction

Journal of Taibah University for Science

Our aim in this article is to present a collocation method to solve two population models for sin... more Our aim in this article is to present a collocation method to solve two population models for single and interacting species. For this, logistic growth model and prey-predator model are examined. These models are solved numerically by Pell-Lucas collocation method. The method gives the approximate solutions of these models in form of truncated Pell-Lucas series. By utilizing Pell-Lucas collocation method, non-linear mathematical models are converted to a system of non-linear algebraic equations. This non-linear equation system is solved and the obtained coefficients are the coefficients of the truncated Pell-Lucas serie solution. Furthermore, the residual correction method is used to find better approximate solutions. All results are shown in tables and graphs for different (N, M) values, and additionally the comparisons are made with other methods from. It is seen that the method gives effective results to the presented model problems.

Research paper thumbnail of Bessel collocation approach for approximate solutions of Hantavirus infection model

New Trends in Mathematical Science, Nov 8, 2017

In this study, a collocation method is introduced to find the approximate solutions of Hantavirus... more In this study, a collocation method is introduced to find the approximate solutions of Hantavirus infection model which is a system of nonlinear ordinary differential equations. The method is based on the Bessel functions of the first kind, matrix operations and collocation points. This method converts Hantavirus infection model into a matrix equation in terms of the Bessel functions of first kind, matrix operations and collocation points. The matrix equation corresponds to a system of nonlinear equations with the unknown Bessel coefficients. The reliability and efficiency of the suggested scheme are demonstrated by numerical applications and all numerical calculations have been done by using a program written in Maple.

Research paper thumbnail of Algorithm for the Numerical Solutions of Volterra Population Growth Model with Fractional Order via Haar Wavelet

Contemporary Mathematics

In this paper, a new collocation method based on Haar wavelet is developed for the numerical solu... more In this paper, a new collocation method based on Haar wavelet is developed for the numerical solution ofthe fractional Volterra model (FVM) for population growth of a species in a closed system. In the proposed method the derivative involved in the nonlinear model is approximated using Haar wavelet and the approximate expressions for the unknown function is obtained by the process of integration, the fractional derivative will be considered in the Caputo sense. The technique of residual correction, which aims to reduce the error of the approximate solution by estimating this error, is discussed in some detail. To show the computational efficiency of the proposed method, the residual correction technique are illustrated with an example. The numerical results are compared with existing methods from the literature. The numerical results show that the method is simply applicable, accurate, efficient and robust.

Research paper thumbnail of Pell-Lucas collocation method to solve high-order linear Fredholm-Volterra integro-differential equations and residual correction

TURKISH JOURNAL OF MATHEMATICS

In this article, a collocation method based on Pell-Lucas polynomials is studied to numerically s... more In this article, a collocation method based on Pell-Lucas polynomials is studied to numerically solve higher order linear Fredholm-Volterra integro differential equations (FVIDE). The approximate solutions are assumed in form of the truncated Pell-Lucas polynomial series. By using Pell-Lucas polynomials and relations of their derivatives, the solution form and its derivatives are brought to matrix forms. By applying the collocation method based on equally spaced collocation points, the method reduces the problem to a system of linear algebraic equations. Solution of this system determines the coefficients of assumed solution. Error estimation is made and also a method with the help of the obtained approximate solution is developed that finds approximate solution with better results. Then, the applications are made on five examples to show that the method is successful. In addition, the results are supported by tables and graphs and the comparisons are made with other methods available in the literature. All calculations in this study have been made using codes written in Matlab.