Numerical approximation for the solution of linear sixth order boundary value problems by cubic B-spline

A. Khalid; M. N. Naeem; P. Agarwal; A. Ghaffar; Z. Ullah; S. Jain

doi:10.1186/s13662-019-2385-9

Numerical approximation for the solution of linear sixth order boundary value problems by cubic B-spline

Advances in Difference Equations ◽

10.1186/s13662-019-2385-9 ◽

2019 ◽

Vol 2019 (1) ◽

Cited By ~ 5

Author(s):

A. Khalid ◽

M. N. Naeem ◽

P. Agarwal ◽

A. Ghaffar ◽

Z. Ullah ◽

...

Keyword(s):

Numerical Approximation ◽

Analytic Solution ◽

Linear Equations ◽

Computational Cost ◽

Spline Method ◽

System Of Linear Equations ◽

Approximation Solution ◽

B Spline ◽

Novel Technique ◽

Derivatives Of

AbstractIn the current paper, authors proposed a computational model based on the cubic B-spline method to solve linear 6th order BVPs arising in astrophysics. The prescribed method transforms the boundary problem to a system of linear equations. The algorithm we are going to develop in this paper is not only simply the approximation solution of the 6th order BVPs using cubic B-spline, but it also describes the estimated derivatives of 1st order to 6th order of the analytic solution at the same time. This novel technique has lesser computational cost than numerous other techniques and is second order convergent. To show the efficiency of the proposed method, four numerical examples have been tested. The results are described using error tables and graphs and are compared with the results existing in the literature.

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A Novel Technique to Solve the Fuzzy System of Equations

Mathematics ◽

10.3390/math8050850 ◽

2020 ◽

Vol 8 (5) ◽

pp. 850

Author(s):

Nasser Mikaeilvand ◽

Zahra Noeiaghdam ◽

Samad Noeiaghdam ◽

Juan J. Nieto

Keyword(s):

Linear System ◽

Fuzzy Number ◽

Fuzzy System ◽

Linear Equations ◽

Second Step ◽

Negative Solution ◽

System Of Linear Equations ◽

Vector Solution ◽

Novel Technique ◽

Fuzzy Linear System

The aim of this research is to apply a novel technique based on the embedding method to solve the n × n fuzzy system of linear equations (FSLEs). By using this method, the strong fuzzy number solutions of FSLEs can be obtained in two steps. In the first step, if the created n × n crisp linear system has a non-negative solution, the fuzzy linear system will have a fuzzy number vector solution that will be found in the second step by solving another created n × n crisp linear system. Several theorems have been proved to show that the number of operations by the presented method are less than the number of operations by Friedman and Ezzati’s methods. To show the advantages of this scheme, two applicable algorithms and flowcharts are presented and several numerical examples are solved by applying them. Furthermore, some graphs of the obtained results are demonstrated that show the solutions are fuzzy number vectors.

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HERMITE POLYNOMIAL APPROACH FOR SOLVING SINGULAR PERTURBATED DELAY DIFFERENTIAL EQUATIONS

Journal of Science and Arts ◽

10.46939/j.sci.arts-20.4-a06 ◽

2020 ◽

Vol 20 (4) ◽

pp. 845-854

Author(s):

SUAYIP YUZBASI ◽

NURCAN BAYKUS SAVASANERIL

Keyword(s):

Algebraic System ◽

Hermite Polynomial ◽

Linear Equations ◽

Hermite Polynomials ◽

System Of Linear Equations ◽

Collocation Points ◽

The Matrix ◽

Collocation Approach ◽

Delay Differential ◽

Derivatives Of

In this study, a collocation approach based on the Hermite polyomials is applied to solve the singularly perturbated delay differential eqautions by boundary conditions. By means of the matix relations of the Hermite polynomials and the derivatives of them, main problem is reduced to a matrix equation. And then, collocation points are placed in equation of the matrix. Hence, the singular perturbed problem is transformed into an algebraic system of linear equations. This system is solved and thus the coefficients of the assumed approximate solution are determined. Numerical applications are made for various values of N.

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A Novel Technique for Numerical Approximation of 2 Dimensional Non-Linear Coupled Burgers’ Equations using Uniform Algebraic Hyperbolic (UAH) Tension B-Spline based Differential Quadrature Method

Applied Mathematics & Information Sciences ◽

10.18576/amis/150215 ◽

2021 ◽

Vol 15 (2) ◽

pp. 217-239

Keyword(s):

Numerical Approximation ◽

Differential Quadrature Method ◽

Differential Quadrature ◽

Quadrature Method ◽

B Spline ◽

Burgers Equations ◽

Novel Technique ◽

Non Linear

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A Numerical Solution of Fredholm Integral Equations of the Second Kind Based on Tight Framelets Generated by the Oblique Extension Principle

Symmetry ◽

10.3390/sym11070854 ◽

2019 ◽

Vol 11 (7) ◽

pp. 854 ◽

Cited By ~ 8

Author(s):

Mutaz Mohammad

Keyword(s):

Integral Equation ◽

Numerical Solution ◽

Integral Equations ◽

Linear Equations ◽

Computational Method ◽

Fredholm Integral Equations ◽

Extension Principle ◽

System Of Linear Equations ◽

B Spline ◽

Extension Principles

In this paper, we present a new computational method for solving linear Fredholm integral equations of the second kind, which is based on the use of B-spline quasi-affine tight framelet systems generated by the unitary and oblique extension principles. We convert the integral equation to a system of linear equations. We provide an example of the construction of quasi-affine tight framelet systems. We also give some numerical evidence to illustrate our method. The numerical results confirm that the method is efficient, very effective and accurate.

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A New Scheme Using Cubic B-Spline to Solve Non-Linear Differential Equations Arising in Visco-Elastic Flows and Hydrodynamic Stability Problems

Mathematics ◽

10.3390/math7111078 ◽

2019 ◽

Vol 7 (11) ◽

pp. 1078 ◽

Cited By ~ 2

Author(s):

Asifa Tassaddiq ◽

Aasma Khalid ◽

Muhammad Nawaz Naeem ◽

Abdul Ghaffar ◽

Faheem Khan ◽

...

Keyword(s):

Differential Equations ◽

Linear Equations ◽

Main Idea ◽

Linear Differential Equations ◽

System Of Linear Equations ◽

B Spline ◽

Stability Problems ◽

Non Linear ◽

Linear Differential ◽

Magnetic Stability

This study deals with the numerical solution of the non-linear differential equations (DEs) arising in the study of hydrodynamics and hydro-magnetic stability problems using a new cubic B-spline scheme (CBS). The main idea is that we have modified the boundary value problems (BVPs) to produce a new system of linear equations. The algorithm developed here is not only for the approximation solutions of the 10th order BVPs but also estimate from 1st derivative to 10th derivative of the exact solution as well. Some examples are illustrated to show the feasibility and competence of the proposed scheme.

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Efficient Method to Approximately Solve Retrial Systems with Impatience

Journal of Applied Mathematics ◽

10.1155/2012/186761 ◽

2012 ◽

Vol 2012 ◽

pp. 1-18

Author(s):

Jose Manuel Gimenez-Guzman ◽

M. Jose Domenech-Benlloch ◽

Vicent Pla ◽

Jorge Martinez-Bauset ◽

Vicente Casares-Giner

Keyword(s):

State Space ◽

Decision Process ◽

Analytic Solution ◽

Computational Cost ◽

Continuous Time Markov Chain ◽

Kolmogorov Equations ◽

Novel Technique ◽

Polynomial Extrapolation ◽

Markov Decision ◽

Infinite State

We present a novel technique to solve multiserver retrial systems with impatience. Unfortunately these systems do not present an exact analytic solution, so it is mandatory to resort to approximate techniques. This novel technique does not rely on the numerical solution of the steady-state Kolmogorov equations of the Continuous Time Markov Chain as it is common for this kind of systems but it considers the system in its Markov Decision Process setting. This technique, known as value extrapolation, truncates the infinite state space using a polynomial extrapolation method to approach the states outside the truncated state space. A numerical evaluation is carried out to evaluate this technique and to compare its performance with previous techniques. The obtained results show that value extrapolation greatly outperforms the previous approaches appeared in the literature not only in terms of accuracy but also in terms of computational cost.

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Linear computational cost implicit solver for elliptic problems

Computer Science ◽

10.7494/csci.2020.21.3.3824 ◽

2020 ◽

Vol 21 (3) ◽

Author(s):

Grzegorz Gurgul ◽

Marcin Los ◽

Maciej Paszynski ◽

Victor Calo

Keyword(s):

Heat Transfer ◽

Transfer Problem ◽

Linear Equations ◽

Computational Cost ◽

Basis Functions ◽

Implicit Method ◽

Time Step ◽

B Spline ◽

Spline Basis ◽

Intermediate Time

In this paper, we use the alternating direction method for isogeometric finite elements to simulate implicit dynamics. Namely, we focus on a parabolic problem and use B-spline basis functions in space and an implicit marching method to fully discretize the problem. We introduce intermediate time steps and separate our differential operator into a summation of the blocks, acting along a particular coordinate axis in the intermediate time steps. We show that the resulting stiffness matrix can be represented as a multiplication of two (in 2D) or three (in 3D) multi-diagonal matrices, each one with B-spline basis functions along the particular axis of the spatial system of coordinates. As a result of this algebraic transformations, we get a system of linear equations that can be factorized in linear $O(N)$ computational cost in every time step of the implicit method. We use our method to simulate the heat transfer problem. We demonstrate theoretically and verify numerically that our implicit method is unconditionally stable for heat transfer problems (i.e., parabolic). We conclude our presentation with a discussion on the limitations of the method.

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Approximate Merging of Two Adjacent B-Spline Surfaces Using Least Square Approximation

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.740.619 ◽

2015 ◽

Vol 740 ◽

pp. 619-623

Author(s):

Zi Zhi Lin ◽

Si Hui Shu

Keyword(s):

Linear Equations ◽

Data Communication ◽

Least Square ◽

System Of Linear Equations ◽

B Spline ◽

The Core ◽

Spline Surface ◽

Spline Surfaces ◽

Approximate Merging ◽

Core Problem

The approximate merging of two adjacent B-spline surfaces into a B-spline surface is the core problem in data communication. A novel algorithm is presented in this paper to solve this problem. In this algorithm, we compute the merging error using L2 norm instead of the Euclidean norm, the process of merging is minimizing the approximate error and we only need solve a system of linear equations to get the final merging surface. In order to reduce the merging error, we add a weighed function on objective function to start the next merger; this function adds greater weight on where error is larger. After the next merger, the merging error will be significantly reduced. Finally, some examples are given to demonstrate the effectiveness and validity of the proposed algorithm.

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