scholarly journals A COMPUTATIONAL ALGORITHM FOR THE NUMERICAL SOLUTION OF NONLINEAR FRACTIONAL INTEGRAL EQUATIONS

Fractals ◽  
2021 ◽  
Author(s):  
ROHUL AMIN ◽  
NORAZAK SENU ◽  
MUHAMMAD BILAL HAFEEZ ◽  
NOREEN IZZA ARSHAD ◽  
ALI AHMADIAN ◽  
...  
Keyword(s):  
Integral Equations ◽  
Fractional Integral ◽  
Computational Algorithm ◽  
Haar Wavelet ◽  
Mean Square ◽  
Algebraic Equations ◽  
Collocation Points ◽  
Collocation Technique ◽  
Uniqueness And Existence ◽  
Stability Of The Solution

In this paper, we develop a numerical method for the solution of nonlinear fractional integral equations (NFIEs) based on Haar wavelet collocation technique (HWCT). Under certain conditions, we also prove the uniqueness and existence as well as Hyers–Ulam (HU) stability of the solution. With the help of the mentioned technique, the considered problem is transformed to a system of algebraic equations which is then solved for the required results by using Broyden algorithm. To check the validation and convergence of the proposed technique, some examples are given. For different number of collocation points (CPs), maximum absolute and mean square root errors are computed. The results show that for solving these equations, the HWCT is effective. The convergence rate is also measured for different CPs, which is nearly equal to [Formula: see text].

scholarly journals THEORETICAL AND COMPUTATIONAL RESULTS FOR MIXED TYPE VOLTERRA–FREDHOLM FRACTIONAL INTEGRAL EQUATIONS

Fractals ◽  
2021 ◽  
Author(s):  
ROHUL AMIN ◽  
HUSSAM ALRABAIAH ◽  
IBRAHIM MAHARIQ ◽  
ANWAR ZEB
Keyword(s):  
Integral Equations ◽  
Mixed Type ◽  
Fractional Integral ◽  
Haar Wavelet ◽  
Algebraic Equations ◽  
Elimination Algorithm ◽  
Collocation Technique ◽  
Uniqueness Of The Solution ◽  
Gauss Elimination ◽  
Mean Square Errors

In this paper, we develop a numerical method for the solutions of mixed type Volterra–Fredholm fractional integral equations (FIEs). The proposed algorithm is based on Haar wavelet collocation technique (HWCT). Under certain conditions, we prove the existence and uniqueness of the solution. Also, some stability results are given of Hyers–Ulam (H–U) type. With the help of the HWCT, the considered problem is transformed into a system of algebraic equations which is then solved for the required results by using Gauss elimination algorithm. Some numerical examples for convergence of the proposed technique are taken from the literature. Maximum absolute and root mean square errors are calculated for different collocation points (CPs). The results show that the HWCT is an effective method for solving FIEs. The convergence rate for different CPS is also calculated, which is nearly equal to 2.

scholarly journals Solution of the Systems of Delay Integral Equations in Heterogeneous Data Communication through Haar Wavelet Collocation Approach

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Hualing Wu ◽  
Rohul Amin ◽  
Asmatullah Khan ◽  
Shah Nazir ◽  
Sultan Ahmad
Keyword(s):  
Integral Equations ◽  
Data Communication ◽  
Haar Wavelet ◽  
Heterogeneous Data ◽  
Test Problems ◽  
Linear System Of Equations ◽  
Haar Functions ◽  
Collocation Points ◽  
Collocation Technique ◽  
Gauss Elimination

In this work, the Haar collocation scheme is used for the solution of the class of system of delay integral equations for heterogeneous data communication. The Haar functions are considered for the approximation of unknown function. By substituting collocation points and applying the Haar collocation technique to system of delay integral equations, we have obtained a linear system of equations. For the solution of this system, an algorithm is developed in MATLAB software. The method of Gauss elimination is utilized for the solution of this system. Finally, by using these coefficients, the solution at collocation points is obtained. The convergence of Haar technique is checked on some test problems.

scholarly journals Second-Order Delay Differential Equations to Deal the Experimentation of Internet of Industrial Things via Haar Wavelet Approach

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Yongtao Xuan ◽  
Rohul Amin ◽  
Fakhar Zaman ◽  
Zohaib Khan ◽  
Imad Ullah ◽  
...  
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Numerical Approach ◽  
Haar Wavelet ◽  
Second Order ◽  
Algebraic Equations ◽  
Elimination Algorithm ◽  
Collocation Points ◽  
Delay Differential ◽  
Mean Square Errors

In this article, an efficient numerical approach for the solution of second-order delay differential equations to deal with the experimentation of the Internet of Industrial Things (IIoT) is presented. With the help of the Haar wavelet technique, the considered problem is transformed into a system of algebraic equations which is then solved for the required results by using Gauss elimination algorithm. Some numerical examples for convergence of the proposed technique are taken from the literature. Maximum absolute and root mean square errors are calculated for various collocation points. The results show that the Haar wavelet method is an effective method for solving delay differential equations of second order. The convergence rate is also measured for various collocation points, which is almost equal to 2.

A Numerical Algorithm for Solving Stiff ODEs and DAEs in Multibody Mechanics

1999 ◽  
Author(s):  
Hajrudin Pasic
Keyword(s):  
Algebraic System ◽  
Numerical Solutions ◽  
Solution Procedure ◽  
Differential Algebraic Equations ◽  
Good Precision ◽  
Algebraic Equations ◽  
Constant Matrix ◽  
Fully Implicit ◽  
Collocation Points ◽  
Collocation Technique

Abstract Presented is an algorithm suitable for numerical solutions of multibody mechanics problems. When s-stage fully implicit Runge-Kutta (RK) method is used to solve these problems described by a system of n ordinary differential equations (ODE), solution of the resulting algebraic system requires 2s3 n3 / 3 operations. In this paper we present an efficient algorithm, whose formulation differs from the traditional RK method. The procedure for uncoupling the algebraic system into a block-diagonal matrix with s blocks of size n is derived for any s. In terms of number of multiplications, the algorithm is about s2 / 2 times faster than the original, nondiagonalized system, as well as s2 times in terms of number of additions/multiplications. With s = 3 the method has the same precision and stability property as the well-known RADAU5 algorithm. However, our method is applicable with any s and not only to the explicit ODEs My′ = f(x, y), where M = constant matrix, but also to the general implicit ODEs of the form f (x, y, y′) = 0. In the solution procedure y is assumed to have a form of the algebraic polynomial whose coefficients are found by using the collocation technique. A proper choice of locations of collocation points guarantees good precision/stability properties. If constructed such as to be L-stable, the method may be used for solving differential-algebraic equations (DAEs). The application is illustrated by a constrained planar manipulator problem.

Shifted Legendre spectral collocation technique for solving stochastic Volterra–Fredholm integral equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Mohamed A. Abdelkawy
Keyword(s):  
Brownian Motion ◽  
Error Analysis ◽  
Integral Equations ◽  
Lagrange Interpolation ◽  
Fredholm Integral Equations ◽  
Algebraic Equations ◽  
Spectral Collocation ◽  
Numerical Examples ◽  
Collocation Technique

Abstract This paper addresses a spectral collocation technique to treat the stochastic Volterra–Fredholm integral equations (SVF-IEs). The shifted Legendre–Gauss–Radau collocation (SL-GR-C) method is developed for approximating the FSV-IDEs. The principal target in our technique is to transform the SVF-IEs to a system of algebraic equations. For computational purposes, the Brownian motion W(x) is discretized by Lagrange interpolation. While the integral terms are interpolated by Legendre–Gauss–Lobatto quadrature. Some numerical examples are given to test the accuracy and applicability of our technique. Also, an error analysis is introduced for the proposed method.

Numerical Solution of Nonlinear Volterra-Fredholm Integral Equations Using Haar Wavelet Collocation Method

2017 ◽  
Vol 18 ◽  
pp. 50-59 ◽  
Author(s):  
S.C. Shiralashetti ◽  
R.A. Mundewadi
Keyword(s):  
Integral Equation ◽  
Error Analysis ◽  
Numerical Solution ◽  
Integral Equations ◽  
Collocation Method ◽  
Haar Wavelet ◽  
Fredholm Integral Equations ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Wavelet Collocation Method

In this paper, we present a numerical solution of nonlinear Volterra-Fredholm integral equations using Haar wavelet collocation method. Properties of Haar wavelet and its operational matrices are utilized to convert the integral equation into a system of algebraic equations, solving these equations using MATLAB to compute the Haar coefficients. The numerical results are compared with exact and existing method through error analysis, which shows the efficiency of the technique.

Solving Two-Dimensional Nonlinear Volterra Integral Equations Using Rationalized Haar Functions in the Complex Plane

2020 ◽  
Vol 12 (3) ◽  
pp. 409-415
Author(s):  
Majid Erfanian ◽  
Hamed Zeidabadi ◽  
Rohollah Mehri
Keyword(s):  
Integral Equations ◽  
Order Of Convergence ◽  
Computational Algorithm ◽  
Numerical Solutions ◽  
Haar Wavelet ◽  
Volterra Integral Equations ◽  
Two Dimensional ◽  
Wavelet Method ◽  
Haar Functions ◽  
Banach Theorem

In this work, two-dimensional rational Haar wavelet method has been used to solve the twodimensional Volterra integral equations. By using fixed point Banach theorem we achieved the order of convergence and the rate of convergence is O(n(2q)n). Numerical solutions of three examples are presented by applying a simple and efficient computational algorithm.

scholarly journals Numerical solution of certain Cauchy singular integral equations using a collocation scheme

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Ali Seifi
Keyword(s):  
Integral Equations ◽  
Singular Integral ◽  
Numerical Approximation ◽  
Singular Integral Equations ◽  
Algebraic Equations ◽  
Approximation Solution ◽  
Collocation Technique ◽  
The Matrix ◽  
Cauchy Singular Integral ◽  
Cauchy Singular Integral Equation

Abstract The present study is devoted to developing a computational collocation technique for solving the Cauchy singular integral equation of the second kind (CSIE-2). Although, several studies have investigated the numerical approximation solution of CSIEs, the strong singularity and accuracy of the numerical methods are still two important challenges for these integral equations. In this paper, we focus on the smooth transformation and implementation of Bessel basis polynomials (BBP). The reduction of the CSIEs-2 into a system of algebraic equations with the Gauss–Legendre collocation points simplifies this technique. The technique of performing numerical approximation of the solution is well presented and illustrated in the matrix form. Also, the convergence and error bound associated with the scheme are established. Finally, several experiments show the reliability and numerical efficiency of the proposed scheme in comparison with other methods.

scholarly journals An operational Haar wavelet method for solving fractional Volterra integral equations

2011 ◽  
Vol 21 (3) ◽  
pp. 535-547 ◽  
Author(s):  
Habibollah Saeedi ◽  
Nasibeh Mollahasani ◽  
Mahmoud Moghadam ◽  
Gennady Chuev
Keyword(s):  
Integral Equations ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Haar Wavelet ◽  
Volterra Integral Equations ◽  
Algebraic Equations ◽  
Global Error Bound ◽  
Wavelet Method ◽  
Haar Wavelet Method ◽  
Abel Integral Equations

An operational Haar wavelet method for solving fractional Volterra integral equationsA Haar wavelet operational matrix is applied to fractional integration, which has not been undertaken before. The Haar wavelet approximating method is used to reduce the fractional Volterra and Abel integral equations to a system of algebraic equations. A global error bound is estimated and some numerical examples with smooth, nonsmooth, and singular solutions are considered to demonstrate the validity and applicability of the developed method.

scholarly journals Haar wavelet method for solution of variable order linear fractional integro-differential equations

2022 ◽  
Vol 7 (4) ◽  
pp. 5431-5443
Author(s):  
Rohul Amin ◽  
◽  
Kamal Shah ◽  
Hijaz Ahmad ◽  
Abdul Hamid Ganie ◽  
...  
Keyword(s):  
Differential Equations ◽  
Haar Wavelet ◽  
Variable Order ◽  
Algebraic Equations ◽  
Elimination Algorithm ◽  
Collocation Technique ◽  
Gauss Elimination ◽  
Collocation Scheme ◽  
The Given ◽  
Derivatives Of

<abstract><p>In this paper, we developed a computational Haar collocation scheme for the solution of fractional linear integro-differential equations of variable order. Fractional derivatives of variable order is described in the Caputo sense. The given problem is transformed into a system of algebraic equations using the proposed Haar technique. The results are obtained by solving this system with the Gauss elimination algorithm. Some examples are given to demonstrate the convergence of Haar collocation technique. For different collocation points, maximum absolute and mean square root errors are computed. The results demonstrate that the Haar approach is efficient for solving these equations.</p></abstract>

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