International Journal of Industrial Mathematics, 2013
In this paper, we introduce an iterative algorithm free from second derivative for solving algebr... more In this paper, we introduce an iterative algorithm free from second derivative for solving algebraic nonlinear equations. The analysis of convergence shows that this iterative algorithm has seventh-order convergence. Per iteration of the new algorithm requires three evaluations of the function and two evaluation of its first derivative. Therefore this algorithm has the efficiency index which equals to 1.477. The results obtained using the algorithm presented here show that the iterative algorithm is very effective and convenient for the algebraic nonlinear equations.
In this paper, we presented a numerical method for solving linear Volterra integral equations (LV... more In this paper, we presented a numerical method for solving linear Volterra integral equations (LVIE) which is based on the non-uniform Haar wavelets. By applying this method, the LVIE is reduced to a linear system of algebraic equations which can be solved by direct method. The min advantage of using nonuniform Haar wavelets is that the time of calculation can be adjusted arbitrarily. Also, we presented the error analysis of the proposed method. Furthermore, two examples are included for the demonstrating the convenient capabilities of the new method.
In this paper, first, we propose an iterative method based on quadrature formula for solving two-... more In this paper, first, we propose an iterative method based on quadrature formula for solving two-dimensional linear fuzzy Fredholm integral equations (2DLFFIE). Then, we prove the error estimation of the method. In addition, we show the numerical stability analysis of the method with respect to the choice of the first iteration. Finally, supporting examples are also provided.
In the present study, in the beginning, we prove the existence and uniqueness of the solution of ... more In the present study, in the beginning, we prove the existence and uniqueness of the solution of nonlinear fuzzy Urysohn-Volterra delay integral equations (NFUVDIE). Then, we propose an iterative method and trapezoidal quadrature rule which numerically solve this equation. In addition, we prove the convergence analysis and error estimate of the proposed numerical method by theorem 3. Eventually, we conclude the efficiency of the presented method. Notice that the study of this equation is important since they have broad applications in various engineering sciences. Recently, a number of researchers suggested variant numerical methods for solving of Volterra fuzzy delay integral equations.
In recent years, fuzzy delay integral equation has received a considerable amount of attention. F... more In recent years, fuzzy delay integral equation has received a considerable amount of attention. From the viewpoint of engineering, time delays in controllers may induce complex dynamic behavior and will limit the performance of the active control. Therefore, time delays play an important role in the dynamic behavior of active control systems. In our paper, we introduce an iterative method using trapezoidal quadrature formula for solving two-dimensional nonlinear fuzzy delay integral equations. In addition, the existence and uniqueness of the solution of these equations are proved. Also, the convergence analysis and a criterion to stop of the presented method are investigated. Moreover, we prove numerical stability analysis of the method through choosing the perturbation in the first iteration. Eventually, some numerical examples are provided to demonstrate the applicability of this procedure.
We consider signed graphs, i.e., graphs with positive or negative signs on their edges. The notio... more We consider signed graphs, i.e., graphs with positive or negative signs on their edges. The notion of signed strongly regular graph is recently defined by the author (Signed strongly regular graphs, Proceeding of 48th Annual Iranian Mathematical Conference, 2017). We construct some families of signed strongly regular graphs with only two distinct eigenvalues. The construction is based on the well-known method known as star complement technique.
In this paper, hybrid Bernstein polynomials and block-pulse functions based on the method of succ... more In this paper, hybrid Bernstein polynomials and block-pulse functions based on the method of successive approximations are applied to obtain the approximate solution of nonlinear fuzzy Fredholm integral equations. The main idea of using the proposed method is that fuzzy integral in any iterative process will be reduced to the crisp integration. Some results concerning the error estimate and stability of the numerical method are presented. Numerical examples are introduced to illustrate the effectiveness and simplicity of the present method.
This paper proposes an accurate numerical approach for computing the solution of twodimensional f... more This paper proposes an accurate numerical approach for computing the solution of twodimensional fractional Volterra integral equations. The operational matrices of fractional integration based on the Hybridization of block-pulse and Taylor polynomials are implemented to transform these equations into a system of linear algebraic equations. The error analysis of the proposed method is examined in detail. Numerical results highlight the robustness and accuracy of the proposed strategy.
Advances in Intelligent Systems and Computing, 2021
In this paper, by introducing a class of new orthogonal basis functions (NFs), we propose a numer... more In this paper, by introducing a class of new orthogonal basis functions (NFs), we propose a numerical method to solve the nonlinear Volterra-Fredholm integral equations of the second kind (NVFIE2). To do this, first, we present the operational matrix of integration of NFs. Then, by applying this matrix, NVFIE2 is reduced to a nonlinear system of algebraic equations. Finally, by solving some numerical examples, we show the efficiency of the proposed method.
International Journal of Industrial Mathematics, 2013
In this paper, we introduce an iterative algorithm free from second derivative for solving algebr... more In this paper, we introduce an iterative algorithm free from second derivative for solving algebraic nonlinear equations. The analysis of convergence shows that this iterative algorithm has seventh-order convergence. Per iteration of the new algorithm requires three evaluations of the function and two evaluation of its rst derivative. Therefore this algorithm has the eciency index which equals to 1 .477. The results obtained using the algorithm presented here show that the iterative algorithm is very eective and convenient for the algebraic nonlinear equations.
In this paper we use a computational method based on CAS wavelets for solving nonlinear fractiona... more In this paper we use a computational method based on CAS wavelets for solving nonlinear fractional order Volterra integral equations. We solve particularly Abel equations. An operational matrix of fractional order integration for CAS wavelets is used. Block Pulse Functions (BPFs) and collocation method are employed to derive a general procedure for forming this matrix. The error analysis of proposed numerical scheme is studied theoretically. Finally, comparison of numerical results with exact solution are shown.
In this paper, existence theorems for the fuzzy Volterra-Fredholm integral equations of mixed typ... more In this paper, existence theorems for the fuzzy Volterra-Fredholm integral equations of mixed type (FVFIEMT) involving fuzzy number valued mappings have been investigated. Then, by using Banach's contraction principle, sufficient conditions for the existence of a unique solution of FVFIEMT are given. Finally, illustrative examples are presented to validate the obtained results.
International Journal of Industrial Mathematics, 2017
T he concept of fuzzy integral was initiated by Dubois and Prade [11] and then investigated by Ka... more T he concept of fuzzy integral was initiated by Dubois and Prade [11] and then investigated by Kaleva [21], Goetschel and Voxman [20], Nanda [23] and others. In [33], the Henstock integral of fuzzy-valued functions is defined, while the fuzzy Riemann integral and its numerical integration was investigated byWu in [34]. In [7], the authors introduced some quadrature rules for the integral of fuzzy-number-valued mappings. Kaleva [21] proposed the existence and uniqueness of the solution of fuzzy differential equations using the Banach fixed point principle. Mordeson and Newman (see [22]) started the study of the subject of fuzzy integral equations. The Banach fixed point principle is the powerful tool to investigate of the existence and uniqueness of the solution
Many problems in finance, mechanics, biology, medical, social sciences and other disciplines can ... more Many problems in finance, mechanics, biology, medical, social sciences and other disciplines can be modeled by stochastic integral equations (SIEs). Given the wide range of applications of SIEs, solving these type of equations is a great importance. Clearly, obtaining the analytic solution of SIEs is often either complicated or impossible. Therefore, the development of numerical methods for solving these types of equations is inevitable. Hence, many authors have proposed several numerical approaches for solving these equations. In [11], authors applied triangular functions for solving SIEs. Asgari et al. suggested stochastic operational matrix based on Bernstein polynomials for obtaining numerical solution of nonlinear SIEs [12]. Cheraghi et al. [2], used new basis functions for solving linear stochastic Volterra integral equations. Authors in [1] used stochastic operational matrix based on Haar wavelets for obtaining numerical solution of nonlinear SIEs. To see another methods for ...
International Journal of Industrial Mathematics, 2019
In this paper, firstly, we review approximation of fuzzy functions by fuzzy bicubic spline... more In this paper, firstly, we review approximation of fuzzy functions by fuzzy bicubic splines interpolation and present a new approach based on the two-dimensional fuzzy splines interpolation and iterative method to approximate the solution of two-dimensional linear fuzzy Fredholm integral equation (2DLFFIE). Also, we prove convergence analysis and numerical stability analysis for the proposed numerical algorithm. Finally, by an example, we show the efficiency of the proposed method.
2020 8th Iranian Joint Congress on Fuzzy and intelligent Systems (CFIS), 2020
In the present study, in the beginning, we prove the existence and uniqueness of the solution of ... more In the present study, in the beginning, we prove the existence and uniqueness of the solution of nonlinear fuzzy Urysohn-Volterra delay integral equations (NFUVDIE). Then, we propose an iterative method and trapezoidal quadrature rule which numerically solve this equation. In addition, we prove the convergence analysis and error estimate of the proposed numerical method by theorem 3. Eventually, we conclude the efficiency of the presented method. Notice that the study of this equation is important since they have broad applications in various engineering sciences. Recently, a number of researchers suggested variant numerical methods for solving of Volterra fuzzy delay integral equations.
In this paper, first, we apply the successive approximations method in terms of midpoint quadratu... more In this paper, first, we apply the successive approximations method in terms of midpoint quadrature formula to solve nonlinear fuzzy Fredholm integral equations of the second kind (NFFIE-2). Considering some assumptions, we acquire a new error estimation. Moreover, we prove the convergence of the proposed method. Then, we study the numerical stability of the proposed method with respect to the first iteration choice. Eventually, to demonstrate the accuracy of the suggested method, we present two numerical examples.
In this article, we use a new method based on orthogonal basis functions for the numerical soluti... more In this article, we use a new method based on orthogonal basis functions for the numerical solution of stochastic Volterra integral equations of the second kind (SVIE). By using this method, a SVIE can be reduced to a linear system of algebraic equations. Finally, to show the efficiency of the proposed method, we give two numerical examples.
In this paper, our aim is to provide two hybrid and non-hybrid efficient method based on non-orth... more In this paper, our aim is to provide two hybrid and non-hybrid efficient method based on non-orthogonal Bernoulli polynomials to approximate solution of linear fuzzy Fredholm integral equations. At first, using Bernoulli basis polynomials and also combining them with known block-pulse functions, we convert the fuzzy integral equations to two algebraic systems. The convergence and error estimates of the methods is also given. Finally, we present some illustrative examples and compare the numerical computational results to confirm the theoretical topics and demonstrate the convergence rate of the methods.
In this paper, a new approach to the numerical solution of Volterra-Fredholm integral equations b... more In this paper, a new approach to the numerical solution of Volterra-Fredholm integral equations by using CAS wavelets in combination with the collocation technique is proposed. First, the unknown function is approximated by using CAS wavelets, then the Volterra-Fredholm integral equation is reduced to the linear or nonlinear system of equations. Moreover, the convergence theorem for the proposed method is given. Finally, illustrative examples are included to show the accuracy and the efficiency of the proposed method.
International Journal of Industrial Mathematics, 2013
In this paper, we introduce an iterative algorithm free from second derivative for solving algebr... more In this paper, we introduce an iterative algorithm free from second derivative for solving algebraic nonlinear equations. The analysis of convergence shows that this iterative algorithm has seventh-order convergence. Per iteration of the new algorithm requires three evaluations of the function and two evaluation of its first derivative. Therefore this algorithm has the efficiency index which equals to 1.477. The results obtained using the algorithm presented here show that the iterative algorithm is very effective and convenient for the algebraic nonlinear equations.
In this paper, we presented a numerical method for solving linear Volterra integral equations (LV... more In this paper, we presented a numerical method for solving linear Volterra integral equations (LVIE) which is based on the non-uniform Haar wavelets. By applying this method, the LVIE is reduced to a linear system of algebraic equations which can be solved by direct method. The min advantage of using nonuniform Haar wavelets is that the time of calculation can be adjusted arbitrarily. Also, we presented the error analysis of the proposed method. Furthermore, two examples are included for the demonstrating the convenient capabilities of the new method.
In this paper, first, we propose an iterative method based on quadrature formula for solving two-... more In this paper, first, we propose an iterative method based on quadrature formula for solving two-dimensional linear fuzzy Fredholm integral equations (2DLFFIE). Then, we prove the error estimation of the method. In addition, we show the numerical stability analysis of the method with respect to the choice of the first iteration. Finally, supporting examples are also provided.
In the present study, in the beginning, we prove the existence and uniqueness of the solution of ... more In the present study, in the beginning, we prove the existence and uniqueness of the solution of nonlinear fuzzy Urysohn-Volterra delay integral equations (NFUVDIE). Then, we propose an iterative method and trapezoidal quadrature rule which numerically solve this equation. In addition, we prove the convergence analysis and error estimate of the proposed numerical method by theorem 3. Eventually, we conclude the efficiency of the presented method. Notice that the study of this equation is important since they have broad applications in various engineering sciences. Recently, a number of researchers suggested variant numerical methods for solving of Volterra fuzzy delay integral equations.
In recent years, fuzzy delay integral equation has received a considerable amount of attention. F... more In recent years, fuzzy delay integral equation has received a considerable amount of attention. From the viewpoint of engineering, time delays in controllers may induce complex dynamic behavior and will limit the performance of the active control. Therefore, time delays play an important role in the dynamic behavior of active control systems. In our paper, we introduce an iterative method using trapezoidal quadrature formula for solving two-dimensional nonlinear fuzzy delay integral equations. In addition, the existence and uniqueness of the solution of these equations are proved. Also, the convergence analysis and a criterion to stop of the presented method are investigated. Moreover, we prove numerical stability analysis of the method through choosing the perturbation in the first iteration. Eventually, some numerical examples are provided to demonstrate the applicability of this procedure.
We consider signed graphs, i.e., graphs with positive or negative signs on their edges. The notio... more We consider signed graphs, i.e., graphs with positive or negative signs on their edges. The notion of signed strongly regular graph is recently defined by the author (Signed strongly regular graphs, Proceeding of 48th Annual Iranian Mathematical Conference, 2017). We construct some families of signed strongly regular graphs with only two distinct eigenvalues. The construction is based on the well-known method known as star complement technique.
In this paper, hybrid Bernstein polynomials and block-pulse functions based on the method of succ... more In this paper, hybrid Bernstein polynomials and block-pulse functions based on the method of successive approximations are applied to obtain the approximate solution of nonlinear fuzzy Fredholm integral equations. The main idea of using the proposed method is that fuzzy integral in any iterative process will be reduced to the crisp integration. Some results concerning the error estimate and stability of the numerical method are presented. Numerical examples are introduced to illustrate the effectiveness and simplicity of the present method.
This paper proposes an accurate numerical approach for computing the solution of twodimensional f... more This paper proposes an accurate numerical approach for computing the solution of twodimensional fractional Volterra integral equations. The operational matrices of fractional integration based on the Hybridization of block-pulse and Taylor polynomials are implemented to transform these equations into a system of linear algebraic equations. The error analysis of the proposed method is examined in detail. Numerical results highlight the robustness and accuracy of the proposed strategy.
Advances in Intelligent Systems and Computing, 2021
In this paper, by introducing a class of new orthogonal basis functions (NFs), we propose a numer... more In this paper, by introducing a class of new orthogonal basis functions (NFs), we propose a numerical method to solve the nonlinear Volterra-Fredholm integral equations of the second kind (NVFIE2). To do this, first, we present the operational matrix of integration of NFs. Then, by applying this matrix, NVFIE2 is reduced to a nonlinear system of algebraic equations. Finally, by solving some numerical examples, we show the efficiency of the proposed method.
International Journal of Industrial Mathematics, 2013
In this paper, we introduce an iterative algorithm free from second derivative for solving algebr... more In this paper, we introduce an iterative algorithm free from second derivative for solving algebraic nonlinear equations. The analysis of convergence shows that this iterative algorithm has seventh-order convergence. Per iteration of the new algorithm requires three evaluations of the function and two evaluation of its rst derivative. Therefore this algorithm has the eciency index which equals to 1 .477. The results obtained using the algorithm presented here show that the iterative algorithm is very eective and convenient for the algebraic nonlinear equations.
In this paper we use a computational method based on CAS wavelets for solving nonlinear fractiona... more In this paper we use a computational method based on CAS wavelets for solving nonlinear fractional order Volterra integral equations. We solve particularly Abel equations. An operational matrix of fractional order integration for CAS wavelets is used. Block Pulse Functions (BPFs) and collocation method are employed to derive a general procedure for forming this matrix. The error analysis of proposed numerical scheme is studied theoretically. Finally, comparison of numerical results with exact solution are shown.
In this paper, existence theorems for the fuzzy Volterra-Fredholm integral equations of mixed typ... more In this paper, existence theorems for the fuzzy Volterra-Fredholm integral equations of mixed type (FVFIEMT) involving fuzzy number valued mappings have been investigated. Then, by using Banach's contraction principle, sufficient conditions for the existence of a unique solution of FVFIEMT are given. Finally, illustrative examples are presented to validate the obtained results.
International Journal of Industrial Mathematics, 2017
T he concept of fuzzy integral was initiated by Dubois and Prade [11] and then investigated by Ka... more T he concept of fuzzy integral was initiated by Dubois and Prade [11] and then investigated by Kaleva [21], Goetschel and Voxman [20], Nanda [23] and others. In [33], the Henstock integral of fuzzy-valued functions is defined, while the fuzzy Riemann integral and its numerical integration was investigated byWu in [34]. In [7], the authors introduced some quadrature rules for the integral of fuzzy-number-valued mappings. Kaleva [21] proposed the existence and uniqueness of the solution of fuzzy differential equations using the Banach fixed point principle. Mordeson and Newman (see [22]) started the study of the subject of fuzzy integral equations. The Banach fixed point principle is the powerful tool to investigate of the existence and uniqueness of the solution
Many problems in finance, mechanics, biology, medical, social sciences and other disciplines can ... more Many problems in finance, mechanics, biology, medical, social sciences and other disciplines can be modeled by stochastic integral equations (SIEs). Given the wide range of applications of SIEs, solving these type of equations is a great importance. Clearly, obtaining the analytic solution of SIEs is often either complicated or impossible. Therefore, the development of numerical methods for solving these types of equations is inevitable. Hence, many authors have proposed several numerical approaches for solving these equations. In [11], authors applied triangular functions for solving SIEs. Asgari et al. suggested stochastic operational matrix based on Bernstein polynomials for obtaining numerical solution of nonlinear SIEs [12]. Cheraghi et al. [2], used new basis functions for solving linear stochastic Volterra integral equations. Authors in [1] used stochastic operational matrix based on Haar wavelets for obtaining numerical solution of nonlinear SIEs. To see another methods for ...
International Journal of Industrial Mathematics, 2019
In this paper, firstly, we review approximation of fuzzy functions by fuzzy bicubic spline... more In this paper, firstly, we review approximation of fuzzy functions by fuzzy bicubic splines interpolation and present a new approach based on the two-dimensional fuzzy splines interpolation and iterative method to approximate the solution of two-dimensional linear fuzzy Fredholm integral equation (2DLFFIE). Also, we prove convergence analysis and numerical stability analysis for the proposed numerical algorithm. Finally, by an example, we show the efficiency of the proposed method.
2020 8th Iranian Joint Congress on Fuzzy and intelligent Systems (CFIS), 2020
In the present study, in the beginning, we prove the existence and uniqueness of the solution of ... more In the present study, in the beginning, we prove the existence and uniqueness of the solution of nonlinear fuzzy Urysohn-Volterra delay integral equations (NFUVDIE). Then, we propose an iterative method and trapezoidal quadrature rule which numerically solve this equation. In addition, we prove the convergence analysis and error estimate of the proposed numerical method by theorem 3. Eventually, we conclude the efficiency of the presented method. Notice that the study of this equation is important since they have broad applications in various engineering sciences. Recently, a number of researchers suggested variant numerical methods for solving of Volterra fuzzy delay integral equations.
In this paper, first, we apply the successive approximations method in terms of midpoint quadratu... more In this paper, first, we apply the successive approximations method in terms of midpoint quadrature formula to solve nonlinear fuzzy Fredholm integral equations of the second kind (NFFIE-2). Considering some assumptions, we acquire a new error estimation. Moreover, we prove the convergence of the proposed method. Then, we study the numerical stability of the proposed method with respect to the first iteration choice. Eventually, to demonstrate the accuracy of the suggested method, we present two numerical examples.
In this article, we use a new method based on orthogonal basis functions for the numerical soluti... more In this article, we use a new method based on orthogonal basis functions for the numerical solution of stochastic Volterra integral equations of the second kind (SVIE). By using this method, a SVIE can be reduced to a linear system of algebraic equations. Finally, to show the efficiency of the proposed method, we give two numerical examples.
In this paper, our aim is to provide two hybrid and non-hybrid efficient method based on non-orth... more In this paper, our aim is to provide two hybrid and non-hybrid efficient method based on non-orthogonal Bernoulli polynomials to approximate solution of linear fuzzy Fredholm integral equations. At first, using Bernoulli basis polynomials and also combining them with known block-pulse functions, we convert the fuzzy integral equations to two algebraic systems. The convergence and error estimates of the methods is also given. Finally, we present some illustrative examples and compare the numerical computational results to confirm the theoretical topics and demonstrate the convergence rate of the methods.
In this paper, a new approach to the numerical solution of Volterra-Fredholm integral equations b... more In this paper, a new approach to the numerical solution of Volterra-Fredholm integral equations by using CAS wavelets in combination with the collocation technique is proposed. First, the unknown function is approximated by using CAS wavelets, then the Volterra-Fredholm integral equation is reduced to the linear or nonlinear system of equations. Moreover, the convergence theorem for the proposed method is given. Finally, illustrative examples are included to show the accuracy and the efficiency of the proposed method.
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