scholarly journals THE NUMERICAL SOLUTION BY THE METHOD OF DIRECT INTEGRALS OF DIFFERENTIATION OF EQUATIONS HAVE AN APPLICATION IN THE GAS FILTRATION THEOREM

2020 ◽  
pp. 147-153
Author(s):  
Abdujabbor Abdurazakov ◽  
Nasiba Makhmudova ◽  
Nilufar Mirzamakhmudova
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Error Estimate ◽  
Numerical Solution ◽  
Direct Method ◽  
Gas Filtration ◽  
Sweep Method ◽  
Time Step

On the basis of the direct method and a combination of differential sweep, the article developed a calculated algorithm for solving gas filtration, thereby taking into account the convergence of the approximate solution to the exact one. KEYWORDS: direct method, sweep method, differential equation, time step, convergence, approximate solution, error estimate.

scholarly journals Construction of Benchmark Problems for Solution of Ordinary Differential Equations

1994 ◽  
Vol 1 (5) ◽  
pp. 403-414 ◽  
Author(s):  
Sangchul Lee ◽  
John L. Junkins
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Approximate Solution ◽  
Numerical Solution ◽  
Solution Process ◽  
Optimal Choice ◽  
Benchmark Problems ◽  
Tuning Parameters ◽  
Forcing Function ◽  
The Relationship

An inverse method is introduced to construct benchmark problems for the numerical solution of initial value problems. Benchmark problems constructed in this fashion have a known exact solution, even though analytical solutions are generally not obtainable. The process leading to the exact solution makes use of an initially available approximate numerical solution. A smooth interpolation of the approximate solution is forced to exactly satisfy the differential equation by analytically deriving a small forcing function to absorb all of the errors in the interpolated approximate solution. Using this special case exact solution, it is possible to directly investigate the relationship between global errors of a candidate numerical solution process and the associated tuning parameters for a given code and a given problem. Under the assumption that the original differential equation is well-posed with respect to the small perturbations, we thereby obtain valuable information about the optimal choice of the tuning parameters and the achievable accuracy of the numerical solution. Five illustrative examples are presented.

Numerical Solution to Volterra-Type Integro-Differential Equations of the Second Kinds by Legendre Collocation Method

2014 ◽  
Vol 687-691 ◽  
pp. 1522-1527
Author(s):  
Ting Jing Zhao
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Differential Equations ◽  
Numerical Solution ◽  
Numerical Method ◽  
Collocation Method ◽  
High Accuracy ◽  
Time Step ◽  
Volterra Type ◽  
Legendre Collocation Method

The purpose of this paper is to propose an efficient numerical method for solving Volterra-type integro-differential equation of the second kinds. This method based on Legendre-Gauss-Radau collocation, which is easy to be implemented especially for nonlinear and possesses high accuracy. Also, the method can be done by proceeding in time step by step. Illustrative examples have been discussed to demonstrate the validity and applicability of the technique, and the results have been compared with the exact solution.

scholarly journals ON THE UNIFORM CONVERGENCE OF THE APPROXIMATE SOLUTION OF SINGULAR INTEGRO-DIFFERENTIAL EQUATION OF THE FIRST KIND

2017 ◽  
Vol 19 (6) ◽  
pp. 54-60
Author(s):  
A.V. Ozhegova ◽  
L.E. Hayrullina
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Orthogonal Polynomials ◽  
Uniform Convergence ◽  
Boundary Problem ◽  
Direct Method ◽  
Sufficient Conditions ◽  
Cauchy Kernel ◽  
Uniform Estimates ◽  
Stated Problem

The article presents the study of the boundary problem for singular inte-gro-differential equation of the first kind with Cauchy kernel. The authors introduce the pair of weight spaces to prove the correctness of the stated problem. The article states the sufficient conditions for the convergence of the general direct method, the method of orthogonal polynomials, and as a result uniform estimates for errors of approximate solution.

Numerical and experimental results on the dispersion of a solute in a fluid in lamin/ar flow through a tube

1962 ◽  
Vol 269 (1338) ◽  
pp. 352-367 ◽  
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Approximate Solution ◽  
Numerical Solution ◽  
Circular Tube ◽  
Experimental Results ◽  
Dilute Solution ◽  
Pure Solvent ◽  
Flow Through ◽  
Good Agreement

The longitudinal dispersion of a solute is studied for the process in which pure solvent slowly displaces solution in a circular tube. Good agreement is obtained between a numerical solution of the partial differential equation describing the process and experimental results for water displacing a dilute solution of potassium permanganate. The range of applicability of Sir Geoffrey Taylor’s approximate solution is discussed. An improved approximate solution is presented with a much wider range of validity. This improved approximation is based on the numerical solution of the equation describing the process.

Normalization Bernstein Basis For Solving Fractional Fredholm-Integro Differential Equation

2018 ◽  
pp. 491
Author(s):  
Abdul Khaleq O. Al-Jubory ◽  
Shaymaa Hussain Salih
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Linear System ◽  
Differential Equations ◽  
Bernstein Basis ◽  
Traditional Methods ◽  
Integro Differential Equation

In this work, we employ a new normalization Bernstein basis for solving linear Freadholm of fractional integro-differential equations  nonhomogeneous  of the second type (LFFIDEs). We adopt Petrov-Galerkian method (PGM) to approximate solution of the (LFFIDEs) via normalization Bernstein basis that yields linear system. Some examples are given and their results are shown in tables and figures, the Petrov-Galerkian method (PGM) is very effective and convenient and overcome the difficulty of traditional methods. We solve this problem (LFFIDEs) by the assistance of Matlab10.   

scholarly journals On the Numerical Simulation of HPDEs Using θ-Weighted Scheme and the Galerkin Method

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 78
Author(s):  
Haifa Bin Jebreen ◽  
Fairouz Tchier
Keyword(s):  
Differential Equation ◽  
Galerkin Method ◽  
Linear Ordinary Differential Equation ◽  
Time Interval ◽  
Hyperbolic Partial Differential Equation ◽  
Time Step ◽  
Numerical Examples ◽  
One Dimensional ◽  
The Stability ◽  
The Galerkin Method

Herein, an efficient algorithm is proposed to solve a one-dimensional hyperbolic partial differential equation. To reach an approximate solution, we employ the θ-weighted scheme to discretize the time interval into a finite number of time steps. In each step, we have a linear ordinary differential equation. Applying the Galerkin method based on interpolating scaling functions, we can solve this ODE. Therefore, in each time step, the solution can be found as a continuous function. Stability, consistency, and convergence of the proposed method are investigated. Several numerical examples are devoted to show the accuracy and efficiency of the method and guarantee the validity of the stability, consistency, and convergence analysis.

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