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 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.

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 Efficient Numerical Scheme for the Solution of Tenth Order Boundary Value Problems by the Haar Wavelet Method

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1874
Author(s):  
Rohul Amin ◽  
Kamal Shah ◽  
Imran Khan ◽  
Muhammad Asif ◽  
Mehdi Salimi ◽  
...  
Keyword(s):  
Boundary Value Problems ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Haar Wavelet ◽  
Nonlinear Boundary Value Problems ◽  
Haar Functions ◽  
Collocation Points ◽  
Wavelet Collocation Method ◽  
Mean Square Errors ◽  
Gauss Points

In this paper, an accurate and fast algorithm is developed for the solution of tenth order boundary value problems. The Haar wavelet collocation method is applied to both linear and nonlinear boundary value problems. In this technqiue, the tenth order derivative in boundary value problem is approximated using Haar functions and the process of integration is used to obtain the expression of lower order derivatives and approximate solution for the unknown function. Three linear and two nonlinear examples are taken from literature for checking validation and the convergence of the proposed technique. The maximum absolute and root mean square errors are compared with the exact solution at different collocation and Gauss points. The experimental rate of convergence using different number of collocation points is also calculated, which is nearly equal to 2.

scholarly journals Haar Wavelet Method for the System of Integral Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Hassan A. Zedan ◽  
Eman Alaidarous
Keyword(s):  
Exact Solution ◽  
Numerical Solution ◽  
Integral Equations ◽  
Haar Wavelet ◽  
Fredholm Integral Equations ◽  
Test Problems ◽  
Wavelet Method ◽  
System Of Integral Equations ◽  
Haar Wavelet Method ◽  
Comparison Of The Results

We employed the Haar wavelet method to find numerical solution of the system of Fredholm integral equations (SFIEs) and the system of Volterra integral equations (SVIEs). Five test problems, for which the exact solution is known, are considered. Comparison of the results is obtained by the Haar wavelet method with the exact solution.

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>

scholarly journals Superconvergence of interpolated collocation solutions for weakly singular Volterra integral equations of the second kind

2021 ◽  
Vol 40 (3) ◽  
Author(s):  
Qiumei Huang ◽  
Min Wang
Keyword(s):  
Integral Equations ◽  
Numerical Experiments ◽  
Volterra Integral Equations ◽  
Least Square ◽  
Convergence Order ◽  
Collocation Points ◽  
Weakly Singular ◽  
Interpolation Postprocessing ◽  
Collocation Solution

AbstractIn this paper, we discuss the superconvergence of the “interpolated” collocation solutions for weakly singular Volterra integral equations of the second kind. Based on the collocation solution $$u_h$$ u h , two different interpolation postprocessing approximations of higher accuracy: $$I_{2h}^{2m-1}u_h$$ I 2 h 2 m - 1 u h based on the collocation points and $$I_{2h}^{m}u_h$$ I 2 h m u h based on the least square scheme are constructed, whose convergence order are the same as that of the iterated collocation solution. Such interpolation postprocessing methods are much simpler in computation. We further apply this interpolation postprocessing technique to hybrid collocation solutions and similar results are obtained. Numerical experiments are shown to demonstrate the efficiency of the interpolation postprocessing methods.

scholarly journals A Comparative Study on Haar Wavelet and Hosaya Polynomial for the numerical solution of Fredholm integral equations

2018 ◽  
Vol 3 (2) ◽  
pp. 447-458 ◽  
Author(s):  
S.C. Shiralashetti ◽  
H. S. Ramane ◽  
R.A. Mundewadi ◽  
R.B. Jummannaver
Keyword(s):  
Error Analysis ◽  
Comparative Study ◽  
Numerical Solution ◽  
Integral Equations ◽  
Haar Wavelet ◽  
Fredholm Integral Equations ◽  
Polynomial Method ◽  
Wavelet Method ◽  
Haar Wavelet Method ◽  
Hosoya Polynomial

AbstractIn this paper, a comparative study on Haar wavelet method (HWM) and Hosoya Polynomial method(HPM) for the numerical solution of Fredholm integral equations. Illustrative examples are tested through the error analysis for efficiency. Numerical results are shown in the tables and figures.

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