Parviz Darania

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A method for the numerical solution of the integro-differential equations

2006-11-21

Parviz Darania , Ali Ebadian

Numerical solution of nonlinear Volterra–Fredholm integro-differential equations

2008-08-13

Parviz Darania , Karim Ivaz

Linearization method for solving nonlinear integral equations

2006-01-01

Parviz Darania , Ali Ebadian , A. V. Oskoi

The objective of this paper is to assess both the applicability and the accuracy of linearization method in several problems of general nonlinear integral equations. This method provides piecewise linear … The objective of this paper is to assess both the applicability and the accuracy of linearization method in several problems of general nonlinear integral equations. This method provides piecewise linear integral equations which can be easily integrated. It is shown that the accuracy of linearization method can be substantially improved by employing variable steps which adjust themselves to the solution. This approach can reveal that, under this method, the nonlinear integral equations can be transformed into the linear integral equations which may be integrated using classical methods. Numerical examples are used to illustrate the preciseness and effectiveness of the proposed method.

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High-order collocation methods for nonlinear delay integral equation

2017-06-07

Parviz Darania , S. Pishbin

Development of the Taylor expansion approach for nonlinear integro-differential equations

2006-01-01

Parviz Darania , Ali Ebadian

In this paper, the Taylor expansion approach is developed for initial value problems for nonlinear integro-differential equations.This method transformed nonlinear integro-differential equation to a matrix equation which corresponds to a … In this paper, the Taylor expansion approach is developed for initial value problems for nonlinear integro-differential equations.This method transformed nonlinear integro-differential equation to a matrix equation which corresponds to a system of nonlinear equations with unknown coefficients.Results of approximate solution to test problems are demonstrated.

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New computational method for solving some 2-dimensional nonlinear Volterra integro-differential equations

2010-08-31

Parviz Darania , Jafar Ahmadi Shali , Karim Ivaz

Collocation Method for Nonlinear Volterra-Fredholm Integral Equations

2012-01-01

Jafar Ahmadi Shali , Parviz Darania , Ali Asgar Jodayree Akbarfam

A fully discrete version of a piecewise polynomial collocation method based on new collocation points, is constructed to solve nonlinear Volterra-Fredholm integral equations. In this paper, we obtain existence and … A fully discrete version of a piecewise polynomial collocation method based on new collocation points, is constructed to solve nonlinear Volterra-Fredholm integral equations. In this paper, we obtain existence and uniqueness results and analyze the convergence properties of the collocation method when used to approximate smooth solutions of Volterra- Fredholm integral equations.

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On the RF‐pair operations for the exact solution of some classes of nonlinear Volterra integral equations

2006-01-01

Parviz Darania , M. Hadizadeh

We study the exact solution of some classes of nonlinear integral equations by series of some invertible transformations and R F ‐pair operations. We show that this method applies to … We study the exact solution of some classes of nonlinear integral equations by series of some invertible transformations and R F ‐pair operations. We show that this method applies to several classes of nonlinear Volterra integral equations as well and give some useful invertible transformations for converting these equations into differential equations of Emden‐Fowler type. As a consequence, we analyze the effect of the proposed operations on the exact solution of the transformed equation in order to find the exact solution of the original equation. Some applications of the method are also given. This approach is effective to find a great number of new integrable equations, which thus far, could not be integrated using the classical methods.

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‎Multistep collocation method for nonlinear delay integral equations

2016-06-01

Parviz Darania

‎The main purpose of this paper is to study the numerical solution of nonlinear Volterra integral equations with constant delays, based on the multistep collocation method. These methods for approximating … ‎The main purpose of this paper is to study the numerical solution of nonlinear Volterra integral equations with constant delays, based on the multistep collocation method. These methods for approximating the solution in each subinterval are obtained by fixed number of previous steps and fixed number of collocation points in current and next subintervals. Also, we analyze the convergence of the multistep collocation method when used to approximate smooth solutions of delay integral equations. Finally, numerical results are given showing a marked improvement in comparison with exact solution.

Multistep collocation methods for integral-algebraic equations with non-vanishing delays

2022-09-06

Parviz Darania , S. Pishbin

Numerical analysis of a high order method for nonlinear delay integral equations

2020-01-25

Parviz Darania , Fatemeh Sotoudehmaram

The relation between the Emden–Fowler equation and the nonlinear heat conduction problem with variable transfer coefficient

2005-02-17

A. R. Zokayi , M. Hadizadeh , Parviz Darania +1 more

On the numerical solutions for nonlinear Volterra-Fredholm integral equations

2021-12-17

Parviz Darania , S. Pishbin

In this note, we study a class of multistep collocation methods for the numerical integration of nonlinear Volterra-Fredholm Integral Equations (V-FIEs). The derived method is characterized by a lower triangular … In this note, we study a class of multistep collocation methods for the numerical integration of nonlinear Volterra-Fredholm Integral Equations (V-FIEs). The derived method is characterized by a lower triangular or diagonal coefficient matrix of the nonlinear system for the computation of the stages which, as it is known, can beexploited to get an efficient implementation. Convergence analysis and linear stability estimates are investigated. Finally numerical experiments are given, which confirm our theoretical results.

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Stability analysis of the high-order multistep collocation method for the functional integral equations with constant delays

2022-01-26

Parviz Darania , Saed Pishbin

The results on the stability of recurrences play an important role in the theory of dynamical systems and computer science in connection to the notions of shadowing and controlled chaos. … The results on the stability of recurrences play an important role in the theory of dynamical systems and computer science in connection to the notions of shadowing and controlled chaos. In this paper, stability properties of high-order multistep collocation method for functional integral equations of Volterra integral equations with constant delays type with respect to significant test equations are investigated.

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Superconvergence analysis of multistep collocation method for delay functional integral equations

2016-07-01

Parviz Darania

In this paper, we will present a review of the multistep collocation method for Delay Volterra Integral Equations (DVIEs) from [1] and then, we study the superconvergence analysis of the … In this paper, we will present a review of the multistep collocation method for Delay Volterra Integral Equations (DVIEs) from [1] and then, we study the superconvergence analysis of the multistep collocation method for DVIEs. Some numerical examples are given to confirm our theoretical results.

On the approximate solutions of the fractional neutron point kinetics equations

2020-07-23

Haleh Sadeghi , Parviz Darania

Convergence Analysis of Multi-Step Collocation Methods to Solve Generalized Auto-Convolution Volterra Integral Equations

2023-06-01

Parviz Darania , S. Pishbin , A. Ebadi

Multistep collocation technique implementation for a pantograph-type second-kind Volterra integral equation

2024-01-01

Shireen Obaid Khaleel , Parviz Darania , S. Pishbin +1 more

In this research, we have elaborated high-rate multistep collocation strategies in order to concern with second-type vanishing delay VIEs. Herein, characteristics of uniqueness, existence, and regularity for both numerical and … In this research, we have elaborated high-rate multistep collocation strategies in order to concern with second-type vanishing delay VIEs. Herein, characteristics of uniqueness, existence, and regularity for both numerical and analytical solutions have been shown. To explore the solvability of the system derived from the numerical method, we have defined particular operators and demonstrated that these operators are both compact and bounded. Solvability is studied by means of the innovative compact operator concepts. The concept of convergence has been examined in greater detail, revealing that the convergence of the method is influenced by the spectral radius of the matrix generated according to the collocation parameters in the difference equation resulting from the method's error. Finally, two numerical examples are given to certify our theoretically gained results. Also, since the proposed numerical method is local in nature, it can be compared to other local methods, such as those used in reference [<xref ref-type="bibr" rid="b1">1</xref>]. We will compare our method with [<xref ref-type="bibr" rid="b1">1</xref>] in the last section.

Convergence analysis of product integration method for nonlinear weakly singular Volterra-Fredholm integral equations

2015-06-01

Parviz Darania , Jafar Ahmadi Shali

In this paper, we studied the numerical solution of nonlinear weakly singular Volterra-Fredholm integral equations by using the product integration method. Also, we shall study the convergence behavior of a … In this paper, we studied the numerical solution of nonlinear weakly singular Volterra-Fredholm integral equations by using the product integration method. Also, we shall study the convergence behavior of a fully discrete version of a product integration method for numerical solution of the nonlinear Volterra-Fredholm integral equations. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments.

Piecewise polynomial numerical method for Volterra integral equations of the fourth-kind with constant delay

2023-08-15

S. Pishbin , Parviz Darania

This work studies the fourth-kind integral equation as a mixed system of first and second-kind Volterra integral equations (VIEs) with constant delay. Regularity, smoothing properties and uniqueness of the solution … This work studies the fourth-kind integral equation as a mixed system of first and second-kind Volterra integral equations (VIEs) with constant delay. Regularity, smoothing properties and uniqueness of the solution of this mixed system are obtained by using theorems which give the relevant conditions related to first and second-kind VIEs with delays. A numerical collocation algorithm making use of piecewise polynomials is designed and the global convergence of the proposed numerical method is established. Some typical numerical experiments are also performed which confirm our theoretical result.

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On the linear stability and super convergence of multi step collocation solutions for functional Volterra integral equations

2024-03-14

Parviz Darania , Fatemeh Sotoudehmaram

Abstract This paper expands on a topic that was explored in recent research by P. Darania et al. (2020), in which the multi-step collocation method (MSCM) has been studied for … Abstract This paper expands on a topic that was explored in recent research by P. Darania et al. (2020), in which the multi-step collocation method (MSCM) has been studied for solving FIEs along with their convergence analysis. This study focuses on the analysis of the linear stability and super-convergence of MSCM for a general form of delay function Θ(t), which includes discontinuity points where the solutions exhibit low regularity. In addition, we present the stability region and the super-convergence order with various examples, thereby demonstrating the accuracy and efficiency of this method. Moreover, the super-convergence and stability regions of the MSCM for FIEs with delay, Θ(t) = t − τ , that do not exhibit any points of discontinuity have been investigated in [4]. 2010 Mathematics Subject Classification: 45G05; 45D05; 65L05; 65L20.

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Study of numerical treatment of functional first-kind Volterra integral equations

2024-01-01

Hassanein Falah , Parviz Darania , S. Pishbin

<abstract>First-kind Volterra integral equations have ill-posed nature in comparison to the second-kind of these equations such that a measure of ill-posedness can be described by $ \nu $-smoothing of the … <abstract>First-kind Volterra integral equations have ill-posed nature in comparison to the second-kind of these equations such that a measure of ill-posedness can be described by $ \nu $-smoothing of the integral operator. A comprehensive study of the convergence and super-convergence properties of the piecewise polynomial collocation method for the second-kind Volterra integral equations (VIEs) with constant delay has been given in [<xref ref-type="bibr" rid="b1">1</xref>]. However, convergence analysis of the collocation method for first-kind delay VIEs appears to be a research problem. Here, we investigated the convergence of the collocation solution as a research problem for a first-kind VIE with constant delay. Three test problems have been fairly well-studied for the sake of verifying theoretical achievements in practice.</abstract>

‎An effective computational approach based on‎ ‎new multi-step collocation method for solving integral-algebraic equations‎

2024-06-10

S‎afaa Hussein ‎Kamil , S. Pishbin , Parviz Darania

<title>Abstract</title> ‎Some efficient numerical schemes have been used to solve integral algebraic equations (IAEs)‎. ‎In this article‎, ‎in order to find higher order method with an extensive accuracy‎, ‎new multi-step … <title>Abstract</title> ‎Some efficient numerical schemes have been used to solve integral algebraic equations (IAEs)‎. ‎In this article‎, ‎in order to find higher order method with an extensive accuracy‎, ‎new multi-step collocation method is developed for the numerical solution of IAEs‎. ‎The convergence of the proposed method is investigated and convergence orders are determined‎. ‎Two examples have been thoroughly examined to validate theoretical advancements in practical application‎. ‎Also‎, ‎we apply one-step and multi-step collocation methods to solve IAEs and compare numerical results to‎ ‎show the efficiency and high accuracy of the proposed method‎. AMS subject classifications: 65R20, 45F15.

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High-rate convergent multistep collocation techniques to a first-kind Volterra integral equation along with the proportional vanishing delay

2024-06-15

Aws Mushtaq Mudheher , S. Pishbin , Parviz Darania +1 more

An effective computational approach based on new multi-step collocation method for solving integral-algebraic equations

2024-11-29

Safa Hussein Kamil , S. Pishbin , Parviz Darania

A Class of Higher‐Order Collocation Scheme for the Integral Equation of the Convolution Type

2025-01-01

M. Alsahlanee , Parviz Darania , S. Pishbin

Auto convolution Volterra integral equations (ACVIEs) are the particular form of non‐standard integral equations arising in mathematical modeling processes and the computation of certain functions. In this paper, a novel … Auto convolution Volterra integral equations (ACVIEs) are the particular form of non‐standard integral equations arising in mathematical modeling processes and the computation of certain functions. In this paper, a novel class of multistep collocation methods (NM‐SCMs) is constructed in order to find a higher‐order method is constructed for the numerical solution of this kind of the ACVIEs. We analyze the convergence of this method and determine the order of convergence. Also, 1‐SCMs and M‐SCMs are applied to solve this equation numerically.

Convergence Analysis of Multi-Step Collocation Method to First-Order Volterra Integro-Differential Equation with Non-Vanishing Delay

2025-05-01

Ahmet Ali Eashel , S. Pishbin , Parviz Darania

Generally, solutions to functional equations involving non-vanishing delays tend to exhibit lower regularity compared to those of smooth functions. In this context, we examine a first-order Volterra integro-differential equation (VIDE) … Generally, solutions to functional equations involving non-vanishing delays tend to exhibit lower regularity compared to those of smooth functions. In this context, we examine a first-order Volterra integro-differential equation (VIDE) with a non-vanishing delay, delving into the characteristics of its solutions. To enhance the accuracy of traditional one-step collocation methods [1], we employ multi-step collocation techniques to obtain numerical solutions for the VIDE with non-vanishing delay. The global convergence properties of the multi-step numerical approach are scrutinized using the Peano Kernel Theorem. Subsequently, for comparative analysis, we utilize a one-step collocation method to numerically solve this equation, showcasing the effectiveness and precision of the multi-step collocation method.

Development of the Nystr\"{o}m Method for Weakly Singular Functional Integral Equations

2025-05-01

Parviz Darania , Bahaa Hussain Alrikabi , S. Pishbin

In this research, we apply the standard product integration method (Nystr\"{o}m method) for solving the delay nonlinear weakly singular Volterra integral equations. Typically, in weakly singular integral equations, the singularity … In this research, we apply the standard product integration method (Nystr\"{o}m method) for solving the delay nonlinear weakly singular Volterra integral equations. Typically, in weakly singular integral equations, the singularity of the kernel leads to the derivatives of the solution becoming singular at the boundary of the domain. The Chelyshkov polynomials serving as orthogonal polynomials, find application in numerical integration. Here, we use roots of these polynomials to make Lagrange interpolating polynomial for approximating the kernel functions in weakly singular functional integral equation. The proposed method's convergence analysis is developed, and numerical examples demonstrate the method's reliability and efficiency.

A semi-analytical-numerical technique to nonlinear stochastic differential equations

2008-12-15

Parviz Darania , Jafar Ahmadi Shali , Karim Ivaz

In this note, we study the approximate solutions of nonlinear stochastic differential equations by using the theories and methods of mathematics analysis. An approximate method based on piecewise linearazation is … In this note, we study the approximate solutions of nonlinear stochastic differential equations by using the theories and methods of mathematics analysis. An approximate method based on piecewise linearazation is developed for the determination of semi-analytical-numerical solution of nonlinear stochastic differential equations. Also, linearazation methods for initial value problems in stochastic differential equations which have singular points are introduced.

Convergence Analysis of Multi-Step Collocation Method to First-Order Volterra Integro-Differential Equation with Non-Vanishing Delay

2025-05-01

Ahmet Ali Eashel , S. Pishbin , Parviz Darania

Development of the Nystr\"{o}m Method for Weakly Singular Functional Integral Equations

2025-05-01

Parviz Darania , Bahaa Hussain Alrikabi , S. Pishbin

A Class of Higher‐Order Collocation Scheme for the Integral Equation of the Convolution Type

2025-01-01

M. Alsahlanee , Parviz Darania , S. Pishbin

An effective computational approach based on new multi-step collocation method for solving integral-algebraic equations

2024-11-29

Safa Hussein Kamil , S. Pishbin , Parviz Darania

High-rate convergent multistep collocation techniques to a first-kind Volterra integral equation along with the proportional vanishing delay

2024-06-15

Aws Mushtaq Mudheher , S. Pishbin , Parviz Darania +1 more

‎An effective computational approach based on‎ ‎new multi-step collocation method for solving integral-algebraic equations‎

2024-06-10

S‎afaa Hussein ‎Kamil , S. Pishbin , Parviz Darania

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On the linear stability and super convergence of multi step collocation solutions for functional Volterra integral equations

2024-03-14

Parviz Darania , Fatemeh Sotoudehmaram

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Study of numerical treatment of functional first-kind Volterra integral equations

2024-01-01

Hassanein Falah , Parviz Darania , S. Pishbin

Multistep collocation technique implementation for a pantograph-type second-kind Volterra integral equation

2024-01-01

Shireen Obaid Khaleel , Parviz Darania , S. Pishbin +1 more

Piecewise polynomial numerical method for Volterra integral equations of the fourth-kind with constant delay

2023-08-15

S. Pishbin , Parviz Darania

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Convergence Analysis of Multi-Step Collocation Methods to Solve Generalized Auto-Convolution Volterra Integral Equations

2023-06-01

Parviz Darania , S. Pishbin , A. Ebadi

Multistep collocation methods for integral-algebraic equations with non-vanishing delays

2022-09-06

Parviz Darania , S. Pishbin

Stability analysis of the high-order multistep collocation method for the functional integral equations with constant delays

2022-01-26

Parviz Darania , Saed Pishbin

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On the numerical solutions for nonlinear Volterra-Fredholm integral equations

2021-12-17

Parviz Darania , S. Pishbin

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On the approximate solutions of the fractional neutron point kinetics equations

2020-07-23

Haleh Sadeghi , Parviz Darania

Numerical analysis of a high order method for nonlinear delay integral equations

2020-01-25

Parviz Darania , Fatemeh Sotoudehmaram

High-order collocation methods for nonlinear delay integral equation

2017-06-07

Parviz Darania , S. Pishbin

Superconvergence analysis of multistep collocation method for delay functional integral equations

2016-07-01

Parviz Darania

‎Multistep collocation method for nonlinear delay integral equations

2016-06-01

Parviz Darania

Convergence analysis of product integration method for nonlinear weakly singular Volterra-Fredholm integral equations

2015-06-01

Parviz Darania , Jafar Ahmadi Shali

Collocation Method for Nonlinear Volterra-Fredholm Integral Equations

2012-01-01

Jafar Ahmadi Shali , Parviz Darania , Ali Asgar Jodayree Akbarfam

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New computational method for solving some 2-dimensional nonlinear Volterra integro-differential equations

2010-08-31

Parviz Darania , Jafar Ahmadi Shali , Karim Ivaz

A semi-analytical-numerical technique to nonlinear stochastic differential equations

2008-12-15

Parviz Darania , Jafar Ahmadi Shali , Karim Ivaz

Numerical solution of nonlinear Volterra–Fredholm integro-differential equations

2008-08-13

Parviz Darania , Karim Ivaz

A method for the numerical solution of the integro-differential equations

2006-11-21

Parviz Darania , Ali Ebadian

On the RF‐pair operations for the exact solution of some classes of nonlinear Volterra integral equations

2006-01-01

Parviz Darania , M. Hadizadeh

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Linearization method for solving nonlinear integral equations

2006-01-01

Parviz Darania , Ali Ebadian , A. V. Oskoi

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Development of the Taylor expansion approach for nonlinear integro-differential equations

2006-01-01

Parviz Darania , Ali Ebadian

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The relation between the Emden–Fowler equation and the nonlinear heat conduction problem with variable transfer coefficient

2005-02-17

A. R. Zokayi , M. Hadizadeh , Parviz Darania +1 more

Coauthor	Papers Together
S. Pishbin	13
Jafar Ahmadi Shali	4
Karim Ivaz	3
Fatemeh Sotoudehmaram	2
Shadi Malek Bagomghaleh	2
M. Hadizadeh	2
Ali Ebadian	2
Saed Pishbin	1
Hassanein Falah	1
S‎afaa Hussein ‎Kamil	1
Aws Mushtaq Mudheher	1
Ali Ebadian	1
Shireen Obaid Khaleel	1
Safa Hussein Kamil	1
M. Alsahlanee	1
Ahmet Ali Eashel	1
Bahaa Hussain Alrikabi	1
Ali Asgar Jodayree Akbarfam	1
A. V. Oskoi	1
Ali M. Rajabi	1
A. Ebadi	1
A. R. Zokayi	1
Haleh Sadeghi	1

Multistep collocation methods for Volterra Integral Equations

2009-01-22

Dajana Conte , Beatrice Paternoster

Collocation methods for Volterra integral and related functional equations.

2006-01-01

I. P. Gavrilyuk

Spectral methods for pantograph-type differential and integral equations with multiple delays

2009-02-13

Ishtiaq Ali , Hermann Brunner , Tao Tang

The numerical solution of weakly singular Volterra functional integro-differential equations with variable delays

2006-01-01

Hermann Brunner

We analyze the attainable order of convergence of collocation solutions forlinear and nonlinear Volterra functional integro-differential equations of neutraltype containing weakly singular kernels and nonvanishing delays. The discretizationof the initial-value … We analyze the attainable order of convergence of collocation solutions forlinear and nonlinear Volterra functional integro-differential equations of neutraltype containing weakly singular kernels and nonvanishing delays. The discretizationof the initial-value problem is based on a reformulation as a sequence of ODEswith nonsmooth solutions. The paper concludes with a brief description of possiblealternative numerical approaches for solving various classes of such functionalintegro-differential equations.

High-order collocation methods for singular Volterra functional equations of neutral type

2006-08-11

Hermann Brunner

Implicitly linear collocation methods for nonlinear Volterra equations

1992-04-01

Hermann Brunner

Iterated collocation methods for Volterra integral equations with delay arguments

1994-01-01

Hermann Brunner

In this paper we give a complete analysis of the global convergence and local superconvergence properties of piecewise polynomial collocation for Volterra integral equations with constant delay. This analysis includes … In this paper we give a complete analysis of the global convergence and local superconvergence properties of piecewise polynomial collocation for Volterra integral equations with constant delay. This analysis includes continuous collocation-based Volterra-Runge-Kutta methods as well as iterated collocation methods and their discretizations.

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High-order collocation methods for nonlinear delay integral equation

2017-06-07

Parviz Darania , S. Pishbin

Multistep collocation approximations to solutions of first-kind Volterra integral equations

2018-04-11

Tingting Zhang , Hui Liang

ON THE CONVERGENCE ANALYSIS OF THE SPLINE COLLOCATION METHOD FOR SYSTEM OF INTEGRAL ALGEBRAIC EQUATIONS OF INDEX-2

2012-12-01

F. Ghoreishi , M. Hadizadeh , S. Pishbin

This article presents some theoretical results for polynomial spline collocation solution to a new class of semi-explicit Integral Algebraic Equations (IAEs) of index-2, which has been introduced in a recent … This article presents some theoretical results for polynomial spline collocation solution to a new class of semi-explicit Integral Algebraic Equations (IAEs) of index-2, which has been introduced in a recent paper of the authors (Hadizadeh, M., Ghoreishi, F. and Pishbin, S. [2011] "Jacobi spectral solution for integral-algebraic equations of index-2, " Appl. Numer. Math.61, 131–148). Critical issues for numerical analysis of the spline collocation method for this type of Volterra systems are discussed and the necessary and sufficient conditions are presented which guarantee the convergence of the method. We analyze the rate of convergence for two disjoint cases of collocation parameter c m . Numerical results confirm the rate of decay of the error predicted by this theory.

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Numerical analysis of a high order method for state-dependent delay integral equations

2013-06-28

Manoochehr Khasi , F. Ghoreishi , M. Hadizadeh

The numerical solution of integral-algebraic equations of index 1 by polynomial spline collocation methods

2000-03-30

J.-P. Kauthen

In this paper, we study polynomial spline collocation methods applied to a particular class of integral-algebraic equations of Volterra type. We analyse mixed systems of second and first kind integral … In this paper, we study polynomial spline collocation methods applied to a particular class of integral-algebraic equations of Volterra type. We analyse mixed systems of second and first kind integral equations. Global convergence and local superconvergence results are established.

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Spline collocation methods for nonlinear Volterra integral equations with unknown delay

1996-07-01

Hermann Brunner , Yuri Yatsenko

Multistep collocation methods for integral-algebraic equations with non-vanishing delays

2022-09-06

Parviz Darania , S. Pishbin

Iterated Collocation Methods for Volterra Integral Equations with Delay Arguments

1994-04-01

Hermann Brunner

In this paper we give a complete analysis of the global convergence and local superconvergence properties of piecewise polynomial collocation for Volterra integral equations with constant delay.This analysis includes continuous … In this paper we give a complete analysis of the global convergence and local superconvergence properties of piecewise polynomial collocation for Volterra integral equations with constant delay.This analysis includes continuous collocation-based Volterra-Runge-Kutta methods as well as iterated collocation methods and their discretizations.

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‎Multistep collocation method for nonlinear delay integral equations

2016-06-01

Parviz Darania

Numerical analysis of a high order method for nonlinear delay integral equations

2020-01-25

Parviz Darania , Fatemeh Sotoudehmaram

Super implicit multistep collocation methods for nonlinear Volterra integral equations

2011-08-29

S. Fazeli , Gholamreza Hojjati , S. Shahmorad

Integral-Algebraic Equations: Theory of Collocation Methods I

2013-01-01

Hui Liang , Hermann Brunner

Our analysis of collocation solutions for general systems of linear integral-algebraic equations (IAEs) is based on the notions of the tractability index and the $\nu$-smoothing property of a Volterra integral … Our analysis of collocation solutions for general systems of linear integral-algebraic equations (IAEs) is based on the notions of the tractability index and the $\nu$-smoothing property of a Volterra integral operator. These are used to decouple the given IAE system into the inherent system of regular second-kind Volterra integral equations (VIEs) and a system of first-kind VIEs. This decoupling is then used to derive the optimal convergence properties of piecewise polynomial collocation solutions. Numerical examples illustrate the theoretical results.

Optimal convergence results of piecewise polynomial collocation solutions for integral–algebraic equations of index-3

2014-11-18

S. Pishbin

Product integration methods for an integral equation with logarithmic singular kernel

1992-04-01

Tao Tang , Sean McKee , Teresa Diogo

Integral-Algebraic Equations: Theory of Collocation Methods II

2016-01-01

Hui Liang , Hermann Brunner

In a previous paper [SIAM J. Numer. Anal., 51 (2013), pp. 2238--2259] we analyzed the optimal orders of convergence of piecewise polynomial collocation solutions for systems of integral-algebraic equations (IAEs) … In a previous paper [SIAM J. Numer. Anal., 51 (2013), pp. 2238--2259] we analyzed the optimal orders of convergence of piecewise polynomial collocation solutions for systems of integral-algebraic equations (IAEs) with tractability index $\mu = 1$. The present paper describes the decoupling of systems of IAEs of tractability index $\mu = 2$ and $(\nu +1)$-smoothing ($\nu \geq 1$). It is then shown that the application of the collocation method to the given system of IAEs is equivalent to the application to the decoupled system. While this in principle forms the basis for an elegant analysis of the optimal order of convergence of the method, we show by an example that collocation is not always feasible for general index-$2$ IAEs. Following the convergence analysis for semiexplicit index-2 IAEs we present two numerical examples: one to verify the predicted orders of convergence and one to show why the collocation method may break down for general IAEs with $\mu = 2$.

On some linear Volterra delay equations

1976-01-01

Jiří Cerha

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Jacobi spectral solution for integral algebraic equations of index-2

2010-08-25

M. Hadizadeh , F. Ghoreishi , S. Pishbin

Convergence analysis of piecewise continuous collocation methods for higher index integral algebraic equations of the Hessenberg type

2013-06-01

Babak Shiri , S. Shahmorad , Gholamreza Hojjati

In this paper, we deal with a system of integral algebraic equations of the Hessenberg type. Using a new index definition, the existence and uniqueness of a solution to this … In this paper, we deal with a system of integral algebraic equations of the Hessenberg type. Using a new index definition, the existence and uniqueness of a solution to this system are studied. The well-known piecewise continuous collocation methods are used to solve this system numerically, and the convergence properties of the perturbed piecewise continuous collocation methods are investigated to obtain the order of convergence for the given numerical methods. Finally, some numerical experiments are provided to support the theoretical results.

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On the convergence of multistep collocation methods for integral-algebraic equations of index 1

2020-10-14

Tingting Zhang , Hui Liang , Shijie Zhang

Collocation methods for integro-differential algebraic equations with index 1

2019-02-18

Hui Liang , Hermann Brunner

Abstract The notion of the tractability index based on the $\nu $-smoothing property of a Volterra integral operator is introduced for general systems of linear integro-differential algebraic equations (IDAEs). It … Abstract The notion of the tractability index based on the $\nu $-smoothing property of a Volterra integral operator is introduced for general systems of linear integro-differential algebraic equations (IDAEs). It is used to decouple the given IDAE system of index $1$ into the inherent system of regular Volterra integro-differential equations (VIDEs) and a system of second-kind Volterra integral equations (VIEs). This decoupling of the given general IDAE forms the basis for the convergence analysis of the two classes of piecewise polynomial collocation methods for solving the given index-$1$ IDAE system. The first one employs the same continuous piecewise polynomial space $S_m^{(0)}$ for both the VIDE part and the second-kind VIE part of the decoupled system. In the second one the VIDE part is discretized in $S_m^{(0)}$, but the second-kind VIE part employs the space of discontinuous piecewise polynomials $S_{m - 1}^{(- 1)}$. The optimal orders of convergence of these collocation methods are derived. For the first method, the collocation solution converges uniformly to the exact solution if and only if the collocation parameters satisfy a certain condition. This condition is no longer necessary for the second method; the collocation solution now converges to the exact solution for any choice of the collocation parameters. Numerical examples illustrate the theoretical results.

On collocation methods for Volterra integral equations with delay arguments

1999-06-20

Vilmoš Horvat

In this paper we construct and give an analysis of the global convergence and local superconvergence properties of polynomial collocation solution u ∈ S m+d(ZN ) of Volterra integral equations … In this paper we construct and give an analysis of the global convergence and local superconvergence properties of polynomial collocation solution u ∈ S m+d(ZN ) of Volterra integral equations with constant delay, thus extending the existing theory for d = −1 to the general case.

Adaptive iterative regularization schemes for two‐dimensional integral‐algebraic systems

2019-07-24

Mahdi S. Farahani , M. Hadizadeh , М. В. Булатов +1 more

The main idea of this paper is to utilize the adaptive iterative schemes based on regularization techniques for moderately ill‐posed problems that are obtained by a system of linear two‐dimensional … The main idea of this paper is to utilize the adaptive iterative schemes based on regularization techniques for moderately ill‐posed problems that are obtained by a system of linear two‐dimensional Volterra integral equations with a singular matrix in the leading part. These problems may arise in the modeling of certain heat conduction processes as well as in the dynamic simulation packages such as compressible flow through a plant piping network. Owing to the ill‐posed nature of the first kind Volterra equation that appears in the system, we will focus on the two families of regularization algorithms, ie, the Landweber and Lavrentiev type methods, where we treat both the exact and perturbed data. Our aim is to work directly with the original Volterra equations without any kind of reduction. Two fast iterative algorithms with reasonable computational complexity are developed. Numerical experiments on a few test problems are used to illustrate the validity and efficiency of the proposed iterative methods in comparison with the classical regularization methods.

Direct regularization for system of integral-algebraic equations of index-1

2017-07-06

Mahdi S. Farahani , M. Hadizadeh

It is known that the coupled system consisting of Volterra integral equations of the first and second kind belongs to the class of moderately ill-posed problems. In the present paper, … It is known that the coupled system consisting of Volterra integral equations of the first and second kind belongs to the class of moderately ill-posed problems. In the present paper, we are interested in numerical solution of Volterra integral-algebraic equations by a direct regularization method, i.e. an approach which does not make use of the adjoint operator as well as any reduction or remodelling of the original problem. A numerical algorithm based on Lavrentiev's regularization iterated method is constructed that preserves the Volterra structure of the original problem. The convergence analysis of the proposed method is given and its validity and efficiency are also demonstrated through several numerical experiments.

Stability analysis of $\theta$ -methods for neutral functional-differential equations

1995-06-01

Yunkang Li

Operational Tau method for singular system of Volterra integro-differential equations

2016-07-29

S. Pishbin

Structural analysis of linear integral-algebraic equations

2019-01-04

Reza Zolfaghari , Nedialko S. Nedialkov

A Hermite-Type Collocation Method for the Solution of an Integral Equation with a Certain Weakly Singular Kernel

1991-01-01

Teresa Diogo , Sean McKee , Tao Tang

A Hermite-type collocation method is considered for the solution of a second-kind Volterra integral equation with a certain weakly singular kernel. The constructed approximation is a cubic spline in the … A Hermite-type collocation method is considered for the solution of a second-kind Volterra integral equation with a certain weakly singular kernel. The constructed approximation is a cubic spline in the continuity class C1. The method is shown to be convergent of order four. Some numerical examples are included.

Numerical solution of integral-algebraic equations for multistep methods

2012-05-01

О. С. Будникова , М. В. Булатов

Taylor polynomial solutions of Volterra integral equations

1994-09-01

Mehmet Sezer

The method of Kanwal and Liu for the solution of Fredholm integral equations is applied to certain linear and nonlinear Volterra integral equations of the second kind. Some equations considered … The method of Kanwal and Liu for the solution of Fredholm integral equations is applied to certain linear and nonlinear Volterra integral equations of the second kind. Some equations considered by other authors are solved in terms of Taylor polynomials and the results are compared.

Numerical solution and structural analysis of two-dimensional integral-algebraic equations

2016-01-11

S. Pishbin

The semi-explicit Volterra integral algebraic equations with weakly singular kernels: The numerical treatments

2012-12-27

S. Pishbin , F. Ghoreishi , M. Hadizadeh

Order of Convergence of One-Step Methods for Volterra Integral Equations of the Second Kind

1983-06-01

Ernst Hairer , Ch. Lubich , S. P. Nørsett

Nice proofs of convergence and asymptotic expansions are known for one-step methods for ordinary differential equations. It is shown that these proofs can be generalized in a natural way to … Nice proofs of convergence and asymptotic expansions are known for one-step methods for ordinary differential equations. It is shown that these proofs can be generalized in a natural way to "extended" one-step methods for Volterra integral equations of the second kind. Furthermore, the convergence of "mixed" one-step methods is investigated. For both types general Volterra–Runge–Kutta methods are considered as examples.

Two-dimensional integral–algebraic systems: Analysis and computational methods

2011-06-26

М. В. Булатов , Pedro M. Lima

The approximate solution of high-order linear Volterra–Fredholm integro-differential equations in terms of Taylor polynomials

2000-06-01

Salih Yalçınbaş , Mehmet Sezer

Convergence analysis of the product integration method for solving the fourth kind integral equations with weakly singular kernels

2020-03-04

Sayed Arsalan Sajjadi , S. Pishbin

A comparison of Adomian's decomposition method and wavelet-Galerkin method for solving integro-differential equations

2002-10-28

Salah M. El‐Sayed , Mohammedi R. Abdel‐Aziz

Numerical solution of nonlinear Volterra–Fredholm integro-differential equations

2008-08-13

Parviz Darania , Karim Ivaz

Taylor polynomial solutions of nonlinear Volterra–Fredholm integral equations

2002-04-01

Salih Yalçınbaş

Convergence and stability of quadrature methods applied to Volterra equations with delay

1993-01-01

Christopher Baker , Mir S. Derakhshan

A class of quadrature methods for the numerical solution of equations of the form y(t)=f(t)+∫t-τtK(t,s,y(s))ds,t∈(0,∞);y(t)=ψ(t),t∈[-τ,0] is proposed and the convergence and stability of the methods are analyzed. The quadrature methods … A class of quadrature methods for the numerical solution of equations of the form y(t)=f(t)+∫t-τtK(t,s,y(s))ds,t∈(0,∞);y(t)=ψ(t),t∈[-τ,0] is proposed and the convergence and stability of the methods are analyzed. The quadrature methods provide numerical approximations which converge under mild assumptions to the true solution on every subset [0 T] of R+ at a rate determined by the local truncation error. The dominant form of the error is established, and the weak stability of a class of formulae follows. Stability restrictions on the choice of discretization parameter are investigated in the case of some linear test equations. Various results are compared and contrasted with those which obtain in the treatment of other evolutionary problems including classical Volterra equations.

On the Generalized Emden–Fowler Equation

1975-04-01

James S. W. Wong

This report contains a survey on recent results concerning solutions of the second order nonlinear differential equation $x'' + a(t)|x|^\gamma \operatorname{sgn} x = 0$, $\gamma > 0$, where $a(t)$ is … This report contains a survey on recent results concerning solutions of the second order nonlinear differential equation $x'' + a(t)|x|^\gamma \operatorname{sgn} x = 0$, $\gamma > 0$, where $a(t)$ is continuous and nonnegative. Specifically, the questions concerning continuability, boundedness, stability, oscillation, asymptotic growth and boundary value problems are discussed. An extensive list of related references is also included.

A method for the approximate solution of the second‐order linear differential equations in terms of Taylor polynomials

1996-11-01

Mehmet Sezer

A matrix method is introduced for the approximate solution of the second‐order linear differential equation with specified associated conditions in terms of Taylor polynomials about any point. Examples are presented … A matrix method is introduced for the approximate solution of the second‐order linear differential equation with specified associated conditions in terms of Taylor polynomials about any point. Examples are presented which illustrate the pertinent features of the method. Also it is applied to the generalized Hermite, Laguerre, Legendre and Chebyshev equations given by Costa and Levine.

Group Analysis of Differential Equations.

1985-02-01

Willard Miller , Л.В. Овсянников

Differential Algebraic Equations, Indices, and Integral Algebraic Equations

1990-12-01

C. W. Gear

In a recent report, Hairer, Lubich, and Roche [Report CH-1211, Dept. de Mathematiques, Universite de Geneve, Geneve, Switzerland, 1988] define the index of differential algebraic equations (DAEs) by considering the … In a recent report, Hairer, Lubich, and Roche [Report CH-1211, Dept. de Mathematiques, Universite de Geneve, Geneve, Switzerland, 1988] define the index of differential algebraic equations (DAEs) by considering the effect of perturbations of the equations on the solutions. This index, which will be called the perturbation index $p_i $, is one more than the number of derivatives of the perturbation that must appear in any estimate of the bound of the change in the solution. An earlier form of index used by a number of authors including Gear and Petzold [SIAM J. Numer. Anal., 21 (1984), pp. 716–728] is determined by the number of differentiations of the DAEs that are required to generate an ordinary differential equation (ODE) satisfied by the solution. This will be called the differential index $d_i $. Hairer, Lubich, and Roche give an example whose differential index is one and perturbation index is two and other examples where they are identical. It will be shown that $d_i \leqq p_i \leqq d_i + 1$ and that $d_i = p_i $ if the derivative components of the DAE are total differentials. This means that the differential components have a first integral. The integrals are a special case of a new type of integral equation which will be called integral algebraic equations (IAEs). (An initial investigation of these equations indicates that they have properties very similar to DAEs.) The converse, namely that $p_i = d_i + 1$ if the differential components are not total differentials, is not true because the system could be composed of a combination of systems of different indices of which the highest index system is a total differential for which $p_i = d_i $, while lower index systems violate the total differential condition without changing the indices of the combined system.