scholarly journals Solution of the Falkner–Skan Equation Using the Chebyshev Series in Matrix Form

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Abdelrady Okasha Elnady ◽  
M. Fayek Abd Rabbo ◽  
Hani M. Negm
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Numerical Method ◽  
Matrix Representation ◽  
Point Of View ◽  
Finite Domain ◽  
Algebraic Equations ◽  
Chebyshev Series ◽  
Infinite Domain ◽  
Computational Point

A numerical method for the solution of the Falkner–Skan equation, which is a nonlinear differential equation, is presented. The method has been derived by truncating the semi-infinite domain of the problem to a finite domain and then expanding the required approximate solution as the elements of the Chebyshev series. Using matrix representation of a function and their derivatives, the problem is reduced to a system of algebraic equations in a simple way. From the computational point of view, the results are in excellent agreement with those presented in published works.

scholarly journals Solution for the System of Lane–Emden Type Equations Using Chebyshev Polynomials

Mathematics ◽  
2018 ◽  
Vol 6 (10) ◽  
pp. 181
Author(s):  
Yalçın ÖZTÜRK
Keyword(s):  
Approximate Solution ◽  
Nonlinear System ◽  
Numerical Method ◽  
Collocation Method ◽  
Chebyshev Polynomials ◽  
Matrix Equation ◽  
Algebraic Equations ◽  
Chebyshev Series ◽  
Solution Function ◽  
The Given

In this paper, we use the collocation method together with Chebyshev polynomials to solve system of Lane–Emden type (SLE) equations. We first transform the given SLE equation to a matrix equation by means of a truncated Chebyshev series with unknown coefficients. Then, the numerical method reduces each SLE equation to a nonlinear system of algebraic equations. The solution of this matrix equation yields the unknown coefficients of the solution function. Hence, an approximate solution is obtained by means of a truncated Chebyshev series. Also, to show the applicability, usefulness, and accuracy of the method, some examples are solved numerically and the errors of the solutions are compared with existing solutions.

A new Reliable Numerical Algorithm Based on the First Kind of Bessel Functions to Solve Prandtl–Blasius Laminar Viscous Flow over a Semi-Infinite Flat Plate

2012 ◽  
Vol 67 (12) ◽  
pp. 665-673 ◽  
Author(s):  
Kourosh Parand ◽  
Mehran Nikarya ◽  
Jamal Amani Rad ◽  
Fatemeh Baharifard
Keyword(s):  
Viscous Flow ◽  
Flat Plate ◽  
Numerical Algorithm ◽  
Bessel Functions ◽  
Nonlinear Ordinary Differential Equation ◽  
Finite Domain ◽  
Algebraic Equations ◽  
Infinite Domain ◽  
Third Order ◽  
Domain Truncation

In this paper, a new numerical algorithm is introduced to solve the Blasius equation, which is a third-order nonlinear ordinary differential equation arising in the problem of two-dimensional steady state laminar viscous flow over a semi-infinite flat plate. The proposed approach is based on the first kind of Bessel functions collocation method. The first kind of Bessel function is an infinite series, defined on ℝ and is convergent for any x ∊ℝ. In this work, we solve the problem on semi-infinite domain without any domain truncation, variable transformation basis functions or transformation of the domain of the problem to a finite domain. This method reduces the solution of a nonlinear problem to the solution of a system of nonlinear algebraic equations. To illustrate the reliability of this method, we compare the numerical results of the present method with some well-known results in order to show the applicability and efficiency of our method.

scholarly journals A Novel and Efficient Numerical Algorithm for Solving 2D Fredholm Integral Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
H. Bin Jebreen
Keyword(s):  
Approximate Solution ◽  
Numerical Method ◽  
Convergence Analysis ◽  
Integral Equations ◽  
Numerical Algorithm ◽  
Numerical Experiments ◽  
Operational Matrix ◽  
Fredholm Integral Equations ◽  
Scaling Functions ◽  
Algebraic Equations

A novel and efficient numerical method is developed based on interpolating scaling functions to solve 2D Fredholm integral equations (FIE). Using the operational matrix of integral for interpolating scaling functions, FIE reduces to a set of algebraic equations that one can obtain an approximate solution by solving this system. The convergence analysis is investigated, and some numerical experiments confirm the accuracy and validity of the method. To show the ability of the proposed method, we compare it with others.

scholarly journals A Numerical Method for Lane-Emden Equations Using Hybrid Functions and the Collocation Method

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Changqing Yang ◽  
Jianhua Hou
Keyword(s):  
Differential Equation ◽  
Numerical Method ◽  
Collocation Method ◽  
Chebyshev Polynomials ◽  
Initial Value Problems ◽  
Algebraic Equations ◽  
Initial Value ◽  
Numerical Examples ◽  
Hybrid Functions ◽  
Block Pulse Functions

A numerical method to solve Lane-Emden equations as singular initial value problems is presented in this work. This method is based on the replacement of unknown functions through a truncated series of hybrid of block-pulse functions and Chebyshev polynomials. The collocation method transforms the differential equation into a system of algebraic equations. It also has application in a wide area of differential equations. Corresponding numerical examples are presented to demonstrate the accuracy of the proposed method.

General formulation of the electric stratified problem with a boundary integral equation

Geophysics ◽  
1995 ◽  
Vol 60 (6) ◽  
pp. 1656-1670 ◽  
Author(s):  
André Straub
Keyword(s):  
Differential Equation ◽  
Kernel Function ◽  
Reference Model ◽  
Boundary Integral Equations ◽  
Hankel Transform ◽  
Cylindrical Symmetry ◽  
Boundary Integral ◽  
Point Of View ◽  
Computational Point ◽  
Linear System Of Equations

The electric potential created by a point source in a stratified model is usually written, in a spectral representation, in terms of a Hankel transform because of the cylindrical symmetry of the model. The solution in the radial wavenumber domain is called the kernel function. This kernel function, as a function of the depth coordinate, is the solution of a 1-D differential equation. The conventional procedure for the calculation of the kernel function consists in applying a recursive scheme. This procedure is effective from a computational point of view but becomes cumbersome from an analytical point of view, especially in the case of an arbitrary number of layers for arbitrary positions of the source and measurement points. I reformulate the problem of the kernel function by establishing the equivalence between the 1-D differential equation and a set of two boundary integral equations. This equivalence lowers the dimension of the problem by one unit so that the integration is performed over a space of dimension zero. The equations thus obtained are called jump summation equations. They are derived from a weighted product of two distinct models. The explicit form of these equations with the use of Green’s kernel (i.e., the kernel function for a homogeneous reference model) leads to the introduction of two basic representations, monopolar and dipolar. Each representation is related to a specific integral operator, but the basic representations are equivalent. The kernel function is computed by solving a linear system of equations. Our formulation is also well adapted to the inverse problem. The relationship between a perturbation of the model and the resulting perturbation of the kernel function is expressed by a Fréchet derivative. This sensitivity quantity is obtained by means of the jump summation equation, and its computation appears straightforward with the basic representations. An application to a novel evaluation of the depth of investigation for usual arrays is given.

Laplace-Weighted Residual Method For Problems with Semi-Infinite Domain

2016 ◽  
Vol 7 (2) ◽  
Author(s):  
Babatunde Sunday Ogundare
Keyword(s):  
Boundary Condition ◽  
Approximate Solution ◽  
Numerical Method ◽  
Laplace Transformation ◽  
Transformation Method ◽  
Weighted Residual Method ◽  
Infinite Domain ◽  
Numerical Examples ◽  
Residual Method ◽  
Laplace Transformation Method

This paper presents an efficient numerical method for the approximate solution of problems involving boundary condition at infinity. The whole idea of the method is based on the combination of Laplace transformation method and weighted residual method. Numerical examples are given to show the validity and applicability of the proposed method and the results obtained are compared with other methods in the literatures.

scholarly journals An approximate solution of one singular integro-differential equation using the method of orthogonal polynomials

2020 ◽  
pp. 86-96
Author(s):  
Galina A. Rasolko ◽  
Sergei M. Sheshko
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Orthogonal Polynomials ◽  
Electromagnetic Waves ◽  
Logarithmic Singularity ◽  
Cauchy Kernel ◽  
Algebraic Equations ◽  
Solution Function ◽  
Integro Differential Equation ◽  
Linear Algebraic Equations

Two computational schemes for solving boundary value problems for a singular integro-differential equation, which describes the scattering of H-polarized electromagnetic waves by a screen with a curved boundary, are constructed.  This equation contains three types of integrals: a singular integral with the Cauchy kernel, integrals with a logarithmic singularity and with the Helder type kernel. The integrands, along with the solution function, contain its first derivative.  The proposed schemes for an approximate solution of the problem are based on the representation of the solution function in the form of a linear combination of the Chebyshev orthogonal polynomials and spectral relations that allows to obtain simple analytical expressions for the singular component of the equation. The expansion coefficients of the solution in terms of the Chebyshev polynomial basis are calculated by solving a system of linear algebraic equations. The results of numerical experiments show that on a grid of 20 –30 points, the error of the approximate solution reaches the minimum limit due to the error in representing real floating-point numbers.

scholarly journals An Efficient Algorithm for Solving Integro-Differential Equation

2019 ◽  
Vol 54 (6) ◽  
Author(s):  
Hassan Hamad AL-Nasrawy ◽  
Abdul Khaleq O. Al-Jubory ◽  
Kasim Abbas Hussaina
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Difference Equations ◽  
Matrix Equation ◽  
Bernstein Polynomials ◽  
Approximate Solutions ◽  
Algebraic Equations ◽  
Approximate Methods ◽  
Linear Algebraic Equations ◽  
Linear Delay

In this paper, we study and modify an approximate method as well as a new collocation method, which is based on orthonormal Bernstein polynomials to find approximate solutions of mixed linear delay Fredholm integro-differential-difference equations under the mixed conditions. The main purpose of this paper is to study and develop some approximate methods to solve the mixed linear delay Fredholm integro-differential-difference equations. We employ a new algorithm to find approximate solution via perpendicular Bernstein polynomials on the interval [0,1], and we construct a new matrix of derivatives that will be used to find an approximate solution of matrix equation, that will reduce it to the systems of linear algebraic equations. We study the convergence approximate solutions to the exact solutions. Finally, two examples are given and their results are shown in figures to illustrate the efficiency and accuracy of this method. All the computations are implemented using Math14.

Numerical solution of Riccati equation using operational matrix method with Chebyshev polynomials

2015 ◽  
Vol 08 (02) ◽  
pp. 1550020
Author(s):  
Yalçın Öztürk ◽  
Mustafa Gülsu
Keyword(s):  
Approximate Solution ◽  
Riccati Equation ◽  
Chebyshev Polynomials ◽  
Matrix Method ◽  
Numerical Technique ◽  
Operational Matrix ◽  
Algebraic Equations ◽  
Chebyshev Series ◽  
Numerical Examples ◽  
Operational Matrix Method

In this paper, we present numerical technique for solving the Riccati equation by using operational matrix method with Chebyshev polynomials. The method consists of expanding the required approximate solution as truncated Chebyshev series. Using operational matrix method, we reduce the problem to a set of algebraic equations. Some numerical examples are given to demonstrate the validity and applicability of the method. The method is easy to implement and produces very accurate results.

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