# Rostam K SaeedSalahaddin University - Erbil | SUH · Department of Mathematics

Rostam K Saeed

Professor

Volterra-Fredholm Integral Equations; Inverse Problem; Numerical solution of PDE

## About

97

Publications

119,398

Reads

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449

Citations

Introduction

Investigation of Different Kinds of Volterra-Fredholm Integral Equations by Different Spline Functions ; iterative methods for solving System of nonlinear equations and Bubble dynamics

Additional affiliations

August 1991 - present

**Salahaddin University - Hawler-College of Science**

Position

- Professaor of Numerical Analysis

Description

- Calculus, Linear Algebra, Integral equation, Approximation theory, Numerical Analysis, Functional Analysis

August 1991 - present

August 1991 - present

**College of Science**

Position

- Academic staff

## Publications

Publications (97)

This study numerically investigates the behaviour of an ultrasonically driven gas bubble between two parallel rigid circular walls with a cylindrical micro-indentation in one wall using the OpenFOAM software. The primary objective is to determine the conditions that facilitate the removal of particulate contamination from the indentation using the...

This study aims to provide an understanding of well-posed and incorrectly-posed problems, as well as the developed methods for solving incorrectly-posed applied problems in mathematics. The history and significance of incorrectly-posed problems in solving various applied problems in the natural sciences are explored in detail. The study of methods...

Solving Volterra-Fredholm integral equations by natural cubic spline function Using the natural cubic spline function, this paper finds the numerical solution of Volterra-Fredholm integral equations of the second kind. The proposed method is based on employing the natural cubic spline function of the unknown function at an arbitrary point and using...

In this paper, when compared to the normal Adomian decomposition approach, we updated the method of calculating Adomian's polynomial to discover the numerical solution for a non-linear coupled Hirota system (CHS) with fewer components, improved accuracy, and faster convergence (ADM). The novel algorithm offers a viable way to computing Adomian poly...

This study determines the numerical solution of linear mixed Volterra-Fredholm integral equations of the second kind using the linear spline function. The proposed method is based on using the unknown function's linear spline function at an arbitrary point and converting the Volterra-Fredholm integral equation into a system of linear equations with...

This study determines the numerical solution of linear mixed
Volterra-Fredholm integral equations of the second kind using the linear spline function. The proposed method is based on using the unknown function’s linear spline function at an arbitrary point and converting the Volterra-Fredholm integral equation into a system of linear equations with...

Using the quadratic spline function, this paper finds the numerical
solution of mixed Volterra-Fredholm integral equations of the second kind. The proposed method is based on employing the quadratic spline function of the unknown function at an arbitrary point and using the integration method to turn the Volterra-Fredholm integral equation into a...

Using the quadratic spline function, this paper finds the numerical solution of mixed Volterra-Fredholm integral equations of the second kind. The proposed method is based on employing the quadratic spline function of the unknown function at an arbitrary point and using the integration method to turn the Volterra-Fredholm integral equation into a s...

Abstract. Data completion known as Cauchy problem is one most investigated inverse problems. In this work we consider a Cauchy problem associated
with Helmholtz equation. Our concerned is the convergence of the well-known
alternating iterative method [25]. Our main result is to restore the convergence for the classical iterative algorithm (KMF) whe...

Understanding the near boundary acoustic oscillation of microbubbles is critical for the effective design of ultrasonic biomedical devices and surface cleaning technologies. Accordingly, this study investigates the three-dimensional microbubble oscillation between two curved rigid plates experiencing a planar acoustic field using boundary integral...

For the computation of a matrix sign function, a family of iterative methods is proposed. for some cases of the accelerator parameters, this method is proven to converge globally with higher rates of convergence. Its stability is discussed analytically as well. to illustrate the efficiency of the proposed methods from the family, several tests of v...

In this paper, two relaxation algorithms on the Dirichlet Neumann boundary condition, for solving the Cauchy problem governed to the Modified Helmholtz equation are presented and compared to the classical alternating iterative algorithm. The numerical results obtained using our relaxed algorithm and the finite element approximation show the numeric...

Dams serving multiple purposes are gaining importance, especially for developing countries as they can provide several enhancement benefits from a single investment has been shown in [9]. So this study inspects the everyday inflow and outflow of water in Mettur Dam from June 2004 to May 2005 and exhibits that whether or not this water benefits agri...

Water renders goods (i.e. drinking water, irrigation water) and services (hydroelectricity generation, amenity) that are being utilized in agriculture, industry and households. Irrigation is a life sustaining component of agriculture production in many developing countries. Agriculture is acknowledged to be the principal utilizer of water all throu...

High speed liquid jet forms when a bubble collapses near a solid boundary. The formation of the jet has both advantages and disadvantages as it causes erosion and damage in the nearby turbomachinery and can be beneficially applied in surface cleaning, fluid pumping, etc. In this paper, three-dimensional bubble oscillation between two curved rigid p...

Maltreatment displays a significant stressor in the life of several youths. In this research we addressed dissymmetry between salivary alpha amylase (SAA) and Cortisol response to a social stressor with maltreated and comparative youth. Comparative youth showed notable connections between the SAA and Cortisol responses; maltreated youth had no nota...

By using the generalized exponential rational function method, we construct the analytical solutions of the mitigating internet bottleneck with quadratic-cubic nonlinearity involving the β-derivative. This equation is described to control internet traffic. A number of new optical soliton solution for them are calculated. Oblique optical solutions a...

An iteration scheme in the class of Steffensen-type methods is proposed and extended to achieve the optimized speed for methods with memory. In fact, 100% convergence acceleration is obtained in contrast to its version without memory and without any additional functional evaluations. Improvements of the convergence radii by this technique are illus...

Steffensen-type methods with memory were originally designed to solve nonlinear equations without the use of additional functional evaluations per computing step. In this paper, a variant of Steffensen’s method is proposed which is derivative-free and with memory. In fact, using an acceleration technique via interpolation polynomials of appropriate...

This is an Open Access Journal / article distributed under the terms of the Creative Commons Attribution License (CC BY-NC-ND 3.0) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. All rights reserved. In this work, we applied Taylor series expansion to find an approximate (s...

In this paper, existence, uniqueness and regularity results are proved for the weak solutions. Furthermore, we try to solve Fisher–KPP equation numerically. Then we compare our results with wave solutions.

In this paper, we apply Radial Basis Function-Pseudospectral method for solving nonlinear Whitham-Broer-Kaup (WBK) shallow water model, numerically. We used radial basis functions to approximate the space derivatives in WBK model for obtaining a system of ordinary differential equations that approximate the WBK model, and fourth-order Runge-Kutta m...

Whitham-Broer-Kaup Shallow Water Model This thesis presents numerical solution of Whitham-Broer-Kaup (WBK) shallow water model using some finite difference methods such as the explicit, implicit, Crank-Nicolson, exponential, Richardson, and DuFort-Frankel methods. The local truncation errors and consistency are also studied for these methods. Furth...

In this paper, we presented finite difference methods for solving nonlinear Whitham-Broer-Kaup (WBK) shallow water model numerically. We first subdivided the domain of the model by a net with a finite number of mesh points, and the derivative at each point replaced by explicit, Crank-Nicolson, and exponential finite difference approximations. The r...

In this paper, we focus on obtaining an approximate solution of the two types of twodimensional linear Volterra-Fredhom integral equations of the second kind. Series
solution method is reformulated and applied with different bases functions for finding
an approximate solution (sometimes the exact solution) for the above two types of
integral equati...

In this paper, Reduced Differential Transform Method (RDTM) has been successively used to find the numerical solutions of the coupled Hirota system (CHS). The results obtained by RDTM are compared with exact solutions to reveal that the RDTM is very accurate and effective. In our work, Maple 13 has been used for computations.

In this paper, we focus on obtaining an approximate solution of the two types of two- dimensional linear Volterra-Fredhom integral equations of the second kind. Series solution method is reformulated and applied with different bases functions for finding an approximate solution (sometimes the exact solution) for the above two types of integral equa...

In this paper, three iteration methods are introduced to solve nonlinear equations. The convergence
criteria for these methods are also discussed. Several examples are presented and compared to other well-known
methods, showing the accuracy and fast convergence of the proposed methods.

After several years as lecture in Numerical Analysis, we felt that the
books that were available on the subject were written in such a way that the students found them difficult to understand. It’s hard for students whose mother language is not English, so we decided that we should try to compose a book on Numerical Analysis that is suitable for st...

We have successfully applied Homotopy analysis method to obtain approximate solution of the
Glycolysis system. Different from perturbation methods, the validity of the HAM is independent on whether or
not there exist small parameters in considered nonlinear equations. Therefore, it provides us with a powerful
analytic tool for strongly nonlinear pr...

In this paper, two methods, namely Homotopy Perturbation Method
(HPM) and Homotopy Analysis Method (HAM) are applied to obtain approximate
solutions of the nonlinear coupled Hirota system (CHS). We see that these two
methods are efficient and effectives for solving nonlinear CHS and the obtained
results of the two methods coincide with each other....

In this paper, a general framework of the Variational iteration method (VIM) is presented for solving coupled Hirota systems (CHS). In VIM, a correction functional is constructed by a general Lagrange's multiplier which can be identified via a Variational theory. Comparison with the exact solutions shows that VIM is a powerful method for the soluti...

The aim of this work is to use the power series expansion with collocation method to approximate the solution of integral
equations (IE) of the second kind on real axis. The technique of this method is based on transforming the IE to a matrix equation which
corresponds to a system of linear equations with unknown coefficients. Two examples are pres...

In this article, we formulate two methods to get approximate solution of Glycolysis system. The first
is Laplace decomposition methods (is a method combined Lplace transform and Adomian polynomial) and
the second is semigroup decomposition method (is a method combined semigroup approach and Adomian
polynomial), In both methods the nonlinear terms i...

In this paper, we found the location and asymptotic of the eigenvalues of the linear differential equation
y
00
+ q(x)y = l
2
p(x)y ; x 2 (0; a)
with the boundary conditions y
0
(a)+ il y(a) = y
0
(0)+ il y(0) = 0 when r (x) > 0 and the normalized condition
R
a
0
r (x)jy(x)j
2
dx = 1 ,
where l is a spectral parameter.

In this paper a new technique of spline methods is used for (0, 1, 3) lacunary interpolation by splines of degree six. An existence and uniqueness theorems of the sextic spline function are studied and also error bound.

In this paper, iterative methods of order three and four constructed based on quadratic spline function for solving nonlinear equations. Several numerical examples are given to illustrate the efficiency and performance of the iterative methods; the methods are also compared with some other known iterative methods.

of nonlinear Fredholm integral equations of the second kind. A comparison
between this approximations and exact solutions for four numerical examples are given,
depending on the least-square error and running computer time. Our numerical results
are obtained by MATLAB 7.0 program and for a system of rank 2×2, 3×3 on Personal
Computer P4.

In this paper, we present new one-and two-steps iterative methods for solving nonlinear equation f(x)=0. It is proved here that the iterative methods converge of order three and six respectively. Several numerical examples are given to illustrate the performance and to show that the iterative methods in this paper give better result than the compar...

n dA ftuuuu dt At η = = 123123123 A Abstract: The aim of this paper is to investigate the performance of the ninth degree spline method for solving the system of ordinary differential equations and to estimate the numerical solution in the whole interval. By considering the maximum absolute errors in the solution at grid points for different choice...

In this paper, we use Romberg algorithm, to find an approximation solution for a
system of nonlinear Fredholm integral equations of the second kind. A comparison
between this approximations and exact solutions for four numerical examples are given,
depending on the least-square error and running computer time. Our numerical results
are obtained...

The Glycolysis model has been solved numerically in one dimension by using two finite differences methods: explicit and Crank-Nicolson method and we were found that the explicit method is simpler while the Crank-Nicolson is more accurate. Also, we found that explicit method is conditionally stable while Crank-Nicolson method is unconditionally stab...

The aim of this paper is to investigate the performance of the ninth degree spline method for solving
the system of ordinary differential equations and to estimate the numerical solution in the whole interval. By
considering the maximum absolute errors in the solution at grid points for different choices of step size, we
conclude that ninth spline...

The aim of this paper is to investigate the performance of the ninth degree spline method for solving
the system of ordinary differential equations and to estimate the numerical solution in the whole interval. By
considering the maximum absolute errors in the solution at grid points for different choices of step size, we
conclude that ninth spline...

The numerical stability analysis of Brusselator system has been done in one and two dimensional space. For one dimension we studied the numerical stability for explicit and implicit (Crank- Nicolson) methods and we found that explicit method for solving Brusselator system is stable under the conditions , and
While the implicit method is uncondition...

In this paper, we present a new third-order iterative method for solving nonlinear equations. The new method is based on Newton-Raphson method and Taylor series method. The efficiency of the method is tested on several numerical examples. It is observed that the method is comparable with the well-known existing methods and in many cases gives bette...

In this paper, we have applied the differential transform method (DTM) to solve systems
of linear or non-linear delay differential equation. A remarkable practical feature of this method its
ability to solve the system of linear or non-linear delay differential equations efficiently. By using
DTM, we manage to obtain the numerical, analytical, and...

Abstract: In this paper, we have applied the differential transform method (DTM) to solve systems
of linear or non-linear delay differential equation. A remarkable practical feature of this method its
ability to solve the system of linear or non-linear delay differential equations efficiently. By using
DTM, we manage to obtain the numerical, analyt...

In this paper, Adomian Decomposition method is applied to solve integral equation systems. If the solution of the system considered as a terms of the series expansion of known functions, then thismethod catches the exact solution.To show capability and robustness of the proposed method, some systems of two-dimensional integral equations are solved.

0,2,5 0,2,5 [1] ] 1 , 0 [ 2 C ] 1 , 0 [ 4 C ABSTRACT The object of this paper is to obtain the existence, uniqueness and upper bounds for errors of six degree spline interpolating the lacunary data (0,2,5). We also showed that the changes of the boundary conditions and the class of spline functions has a main role in minimizing the upper bounds for...

The Brusselator model has been solved numerically in one and two dimensions by using two finite differences methods: For one dimension we used explicit and crank-Nicolson method and we were found that the explicit method is simpler while the Crank-Nicolson is more accurate. For the two dimensions we used the ADE and the ADI methods and we found tha...

In this paper, we consider the construction of the sextic splines
function which interpolating the lacunary data. Also, under suitable
conditions, we show that the existence and uniqueness of the solution.
The convergence analysis of this spline function is studied and the error
bounds are derived. This spline function applied to find an approximat...

In this paper, four types of weighted residual methods (Collocation, Subdomain, Galerkin and least-square methods) are presented for finding an approximate solution of the Brusselator model. We showed the efficiency of the prescribed methods by solving numerical example.

In this article we use Adomian decomposition method (ADM), to solve system of delay differential equations of the first order and delay differential equations of higher order by converting it into a system of delay differential of the first order. Some examples are presented to show the ability of the method for linear and non-linear system of dela...

In this paper, we present a family of new iterative methods for solving nonlinear equations based on Newton's method. The order of convergence and corresponding error equations of the obtained iteration formulae are derived analytically and with the help of M aple. Some numerical examples are given to illustrate the efficiency of the presented meth...

Homotopy perturbation method has been employed to obtain a solution of a system of linear Fredholm fractional integro-differential equations: where denotes Remann -Leiouville fractional derivatives.

In this article we use Adomian decomposition method (ADM), to solve system of delay
differential equations of the first order and delay differential equations of higher order by converting
it into a system of delay differential of the first order. Some examples are presented to show the
ability of the method for linear and non-linear system of dela...

The object of this paper is to obtain the existence, uniqueness and upper bounds for errors of six degree spline interpolating the lacunary data (0,2,5). We also showed that the changes of the boundary conditions and the class of spline functions has a main role in minimizing the upper bounds for error in lacunary interpolation problem. For this re...

In this paper, we consider the construction of the sextic splines function which interpolating the lacunary data. Also, under suitable conditions, we show that the existence and uniqueness of the solution. The convergence analysis of this spline function is studied and the error bounds are derived. This spline function applied to find an approximat...

The aim of this work is to construct lacunary interpolation based
on quartic C3-spline and to apply this spline function for finding
approximate values of smooth function and its continuous derivatives.
Upper bounds for errors and convergence analysis of the presented
lacunary interpolation studied. Also, we have solved numerically two
examples, to...

In this paper two-dimensional quadrature methods are applied to find the approximate solution for a system of two-dimensional linear Fredholm integral equation of the second kind (SL2DFIE). A reliable MATLAB program for solving SL2DFIE was established. Some illustrative examples and comparison tables depending on the least square error are presente...

Problem statement: The lacunary interpolation problem, which we had investigated in this study, consisted in finding the six degree spline S(x) of deficiency four, interpolating data given on thefunction value and third and fifth order in the interval [0,1]. Also, an extra initial condition was prescribed on the first derivative. Other purpose of t...

In this paper, three types of weighted residual methods (Collocation, Subdomain, and Galerkin methods) are presented for finding an approximate (sometimes exact) solution of the system of non-linear Volterra integral equations of the second kind (VIEK2). We showed the efficiency of the prescribed methods by solving some numerical examples.

this paper, we suggest a new iterative method for solving nonlinear equation f(x)=0. It is established that the proposed method has fourth-order convergence. Several numerical examples are given to illustrate the performance of the presented method.

In this paper , a quintic spline interpolation algor ithm presented for the solution o f s e c o nd order
i ni t i a l va l ue problems with a new class of the lacunary spline interpolation based on quintic C -spline s
4
as an approximation to the exact solution of such problems. Convergence analysis of the prese nted spline
function was discussed,...

In this paper, we suggest two new iterative methods for solving nonlinear equations which are improvements of supper-Halley method. It is shown that the proposed methods have fourth and seventh order of convergence respectively. Several numerical examples are given to illustrate the performance of the presented methods. Also show that the iterative...

In this paper, we suggest a new iterative method for solving nonlinear equation f(x)=0. It is established that the proposed method has fourth-order convergence. Several numerical examples are given to illustrate the performance of the presented method.

In this study, a new two-step and three-step iterative methods are derived for solving nonlinear equations. It is shown that the three-step iterative method has fourth-order convergence. Several numerical examples are given to illustrate the efficiency and performance of the new method.

This study deals with introducing spline function to find the approximate solution of the system of nonlinear Fredholm integral equations of the second kind. The benefit of spline functions was demonstrated by presenting several examples.

In this paper, a method for solving linear system of Volterra integral equation of the second kind numerically presented based on Monte-Carlo techniques. Numerical examples illustrate the pertinent features of the method with the proposed system.

The aim of this paper is for finding the numerical solution (sometimes exact) for non-linear system of Volterra integral equations of the second kind (NSVIEK2) by using block-by-block method. Which avoid the need for special starting procedures, but uses numerical quadrature rule. Also some illustrative examples are presented, to elucidate the accu...

In this article, the homotopy perturbation method is applied to solve system of nonlinear Fredholm integral equations of the second kind, and a new computer program by using MATLAB 7.0 was established. This program is very useful for solving (m x m; m $2) a system of nonlinear Fredholm integral equations. Results of the examples indicated that the...

In this paper, we consider the linear system of Fredholm integral equation of the second kind. Three methods are used to solve this system, successive approximation method, Aitken's method depending on successive approximation method and a new procedure which is Aitken's method depending on Adomian decomposition method. A comparison between approxi...

In this paper we consider system of linear Volterra integro-differential equations of the second kind. Two methods are used to solve this system, collocation method and partition method. A comparison between approximate and exact results for two numerical examples depending on the least-square error is given, to show the accuracy of the results obt...

In this paper, we use spectral method for a first time to solve systems of linear Volterra integral and integro-differential equations of the second kind. We use a new idea which is equating the coefficients of the same powers from both sides in each equation. A comparison between approximate and exact results for some numerical examples depending...

The main object in this thesis is to study and modify some numerical
methods to treat two major systems of the Volterra integral equations of
the second kind, which are:
• System of linear Volterra integral equations
• System of linear Volterra integro-differential equations of order n≥1.
Hence, the existence and uniqueness theorem for single linea...

In this paper, we changed the boundary conditions which are given by [6, chapter two] from second derivative to third derivative. Accordingly, it was observed that and we show that the change of the boundary conditions affect in minimizing the error bounds for lacunary interpolation by spline function. Introduction: Recently Jwamer, K. H. [6] obtai...

in this paper , we changed the boundary conditions which are given by [6, chapter two] from second derivative to third derivative . Accordingly , it was observed that we show that the change of boundary conditions affect on minimizing error bounds for lacunary interpolation by spline function.

In this paper we consider for a first time, linearsystem of Fredholm integro-differential equation
of the second kind. Four methods are used to solve this system, successive approximation method,
Aitken's method depending on successive approximation method, Adomian decomposition method and
Aitken's method depending on Adomian decomposition metho...

In this work, we define the joint normal eigenvalue, joint reducing approximate eigenvalue and we obtain some important and new results about the boundary of joint numerical range of n-tuple operators on a complex Hilbert space.

The object of this paper is to obtain the existence, uniqueness and error bounds of quintic deficient splines interpolating the data (0,3).

The object of this paper is to obtain the existence, uniqueness and error bounds of quantic deficient spline interpolating the data (0,3)

The object of this paper is to obtain existence, uniqueness, and error bounds of quintic deficient splines interpolating the data (0,3)

The purpose of this paper is to define joint Hermitian, joint skew-Hermitian and joint quasi nilpotent of n-tuple operators on a complex Hilbert space with some reasonable results concerning the joint numerical range of n-tuple operators on a complex Hilbert space.