scholarly journals Application of Rational Second Kind Chebyshev Functions for System of Integrodifferential Equations on Semi-Infinite Intervals

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
M. Tavassoli Kajani ◽  
S. Vahdati ◽  
Zulkifly Abbas ◽  
Mohammad Maleki
Keyword(s):  
Approximate Solution ◽  
Differential Equations ◽  
Galerkin Method ◽  
High Accuracy ◽  
Integrodifferential Equations ◽  
Expansion Test ◽  
Infinite Intervals ◽  
System Of Integrodifferential Equations ◽  
Rational Chebyshev ◽  
Chebyshev Functions

Rational Chebyshev bases and Galerkin method are used to obtain the approximate solution of a system of high-order integro-differential equations on the interval [0,∞). This method is based on replacement of the unknown functions by their truncated series of rational Chebyshev expansion. Test examples are considered to show the high accuracy, simplicity, and efficiency of this method.

A New Numerical Method for Solving Nonlinear Fractional Fokker–Planck Differential Equations

2017 ◽  
Vol 12 (5) ◽  
Author(s):  
BeiBei Guo ◽  
Wei Jiang ◽  
ChiPing Zhang
Keyword(s):  
Approximate Solution ◽  
Differential Equations ◽  
Reproducing Kernel ◽  
Numerical Solutions ◽  
Simple Algorithm ◽  
High Accuracy ◽  
Computational Work ◽  
Reproducing Kernel Theory ◽  
Transport Problems ◽  
Fokker Planck

The nonlinear fractional-order Fokker–Planck differential equations have been used in many physical transport problems which take place under the influence of an external force filed. Therefore, high-accuracy numerical solutions are always needed. In this article, reproducing kernel theory is used to solve a class of nonlinear fractional Fokker–Planck differential equations. The main characteristic of this approach is that it induces a simple algorithm to get the approximate solution of the equation. At the same time, an effective method for obtaining the approximate solution is established. In addition, some numerical examples are given to demonstrate that our method has lesser computational work and higher precision.

scholarly journals The Laplace-Adomian-Pade Technique for the ENSO Model

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Yi Zeng
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Approximate Solution ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Initial Conditions ◽  
Approximate Solutions ◽  
Decomposition Methods ◽  
High Accuracy ◽  
Padé Method

The Laplace-Adomian-Pade method is used to find approximate solutions of differential equations with initial conditions. The oscillation model of the ENSO is an important nonlinear differential equation which is solved analytically in this study. Compared with the exact solution from other decomposition methods, the approximate solution shows the method’s high accuracy with symbolic computation.

scholarly journals Approximate Solution of Perturbed Volterra-Fredholm Integrodifferential Equations by Chebyshev-Galerkin Method

2017 ◽  
Vol 2017 ◽  
pp. 1-6
Author(s):  
K. Issa ◽  
F. Salehi
Keyword(s):  
Approximate Solution ◽  
Galerkin Method ◽  
Integrodifferential Equation ◽  
Numerical Results ◽  
Integrodifferential Equations ◽  
Right Hand ◽  
The Right

In this work, we obtain the approximate solution for the integrodifferential equations by adding perturbation terms to the right hand side of integrodifferential equation and then solve the resulting equation using Chebyshev-Galerkin method. Details of the method are presented and some numerical results along with absolute errors are given to clarify the method. Where necessary, we made comparison with the results obtained previously in the literature. The results obtained reveal the accuracy of the method presented in this study.

scholarly journals Fast methods for the solution of singular integro-differential and differential equations

1981 ◽  
Vol 4 (4) ◽  
pp. 775-794
Author(s):  
L. F. Abd-Elal
Keyword(s):  
Differential Equations ◽  
Galerkin Method ◽  
Numerical Solution ◽  
Integrodifferential Equations ◽  
Solution Time ◽  
First Order ◽  
Chebyshev Expansion ◽  
Fast Methods ◽  
The Galerkin Method

Uniform methods based on the use of the Galerkin method and different Chebyshev expansion sets are developed for the numerical solution of linear integrodifferential equations of the first order. These methods take a total solution time0(N2lnN)usingNexpansion functions, and also provide error extimates which are cheap to compute. These methods solve both singular and regular integro-differential equations. The methods are also used in solving differential equations.

scholarly journals Solving boundary value problems for partial differential equations in triangular domains by the least squares collocation method

2018 ◽  
pp. 96-111
Author(s):  
В.П. Шапеев ◽  
В.А. Беляев
Keyword(s):  
Approximate Solution ◽  
Partial Differential Equations ◽  
Analytical Solution ◽  
Differential Equations ◽  
Least Squares ◽  
Boundary Value Problems ◽  
Collocation Method ◽  
High Accuracy ◽  
Regular Grid ◽  
Least Squares Collocation

Предложен и реализован новый вариант метода коллокации и наименьших квадратов (КНК) повышенной точности для численного решения краевых задач для уравнений с частными производными (PDE, Partial Differential Equations) в треугольных областях. Реализация этого подхода и численные эксперименты выполнены на примерах решения уравнения Пуассона и бигармонического уравнения. Решение второго уравнения с повышенной точностью использовано для моделирования напряженно-деформированного состояния (НДС) изотропной треугольной пластины, находящейся под действием поперечной нагрузки. Дифференциальные задачи методом КНК проектируются в пространство полиномов четвертой степени. Граничные условия для приближенного решения задач выписываются точно на границе расчетной области, что позволяет теоретически неограниченно повышать порядок точности метода КНК. В новом варианте используются регулярная сетка с прямоугольными ячейками в области решения задачи и на границе области "одинарный" слой нерегулярных ячеек, отсеченных границей от прямоугольных ячеек начальной регулярной сетки. Треугольные нерегулярные граничные ячейки присоединяются к соседним четырехугольным или пятиугольным ячейкам, и в объединенных ячейках строится свой отдельный кусок аналитического решения. При этом в граничных ячейках, которые пересекла граница, для аппроксимации дифференциальных уравнений использованы "законтурные" (расположенные вне расчетной области) точки коллокации и точки согласования решения задачи. Эти два приема позволили существенно уменьшить обусловленность системы линейных алгебраических уравнений приближенной задачи по сравнению со случаем, когда треугольные ячейки использовались как самостоятельные для построения приближенного решения задачи и не была использована "законтурная" часть граничных ячеек. Показано преимущество рассматриваемого подхода перед подходом с применением отображения треугольной области на прямоугольную. В численных экспериментах по анализу сходимости приближенного решения различных задач на последовательности сеток установлено, что решение сходится с повышенным порядком и с высокой точностью совпадает с аналитическим решением задачи в случае, когда оно известно. A high-accuracy new version of the least squares collocation method (LSC) is proposed and implemented for the numerical solution of boundary value problems for PDEs in triangular domains. The implementation of this approach and numerical experiments are performed using the examples of the biharmonic and Poisson equations. The solution of the biharmonic equation with high accuracy is used to simulate the stress-strain state of an isotropic triangular plate under the action of a transverse load. The differential problems are projected onto the space of fourth-degree polynomials by the LSC method. The boundary conditions for the approximate solution are given exactly on the boundary of the computational domain, which allows us theoretically and indefinitely to increase the order of accuracy of the LSC. The new version of the LSC utilizes a regular grid with rectangular cells inside the domain of the solution. It is relatively easy to use a "single" layer of irregular cells that are cut off by the boundary from the rectangular cells of the initial regular grid. Triangular irregular boundary cells are joint to the adjacent quadrangular or pentagonal cells. Thus, a separate piece of the analytical solution is constructed in combined cells. The collocation and matching points situated outside the domain are used to approximate the differential equations in the boundary cells crossed by the boundary. These two methods allows us to reduce significantly the condition number of the system of linear algebraic equations in the approximate compared to the case when the triangular cells are used as independent ones for constructing an approximate solution of the problem and when the extraboundary part of the boundary cells is not used. The advantage of the proposed approach is shown in comparison with the approach using the mapping of the triangular domain onto the rectangular one. It is also shown that the approximate solution converges with a high order and is coincident with the analytical solution of the test problems with a high accuracy.

scholarly journals On the Approximate Solution of Partial Integro-Differential Equations Using the Pseudospectral Method Based on Chebyshev Cardinal Functions

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 286
Author(s):  
Fairouz Tchier ◽  
Ioannis Dassios ◽  
Ferdous Tawfiq ◽  
Lakhdar Ragoub
Keyword(s):  
Approximate Solution ◽  
Differential Equations ◽  
Convergence Analysis ◽  
Efficient Method ◽  
Pseudospectral Method ◽  
Numerical Examples ◽  
Expansion Coefficients ◽  
Cardinal Functions ◽  
Theoretical Results ◽  
Chebyshev Functions

In this paper, we apply the pseudospectral method based on the Chebyshev cardinal function to solve the parabolic partial integro-differential equations (PIDEs). Since these equations play a key role in mathematics, physics, and engineering, finding an appropriate solution is important. We use an efficient method to solve PIDEs, especially for the integral part. Unlike when using Chebyshev functions, when using Chebyshev cardinal functions it is no longer necessary to integrate to find expansion coefficients of a given function. This reduces the computation. The convergence analysis is investigated and some numerical examples guarantee our theoretical results. We compare the presented method with others. The results confirm the efficiency and accuracy of the method.

scholarly journals Modified Laguerre wavelet based Galerkin method for fractional and fractional-order delay differential equations

2019 ◽  
Vol 23 (Suppl. 1) ◽  
pp. 13-21 ◽  
Author(s):  
Aydin Secer ◽  
Neslihan Ozdemir
Keyword(s):  
Approximate Solution ◽  
Integral Operator ◽  
Differential Equations ◽  
Galerkin Method ◽  
Fractional Order ◽  
Delay Differential Equations ◽  
Fractional Integral ◽  
Approximate Solutions ◽  
Delay Differential ◽  
The Given

The application of modified Laguerre wavelet with respect to the given conditions by Galerkin method to an approximate solution of fractional and fractional-order delay differential equations is studied in this paper. For the concept of fractional derivative is used Caputo sense by using Riemann-Liouville fractional integral operator. The presented method here is tested on several problems. The approximate solutions obtained by presented method are compared with the exact solutions and is shown to be a very efficient and powerful tool for obtaining approximate solutions of fractional and fractional-order delay differential equations. Some tables and figures are presented to reveal the performance of the presented method.

scholarly journals A STUDY OF PSEUDO SPECTRAL GALERKIN METHOD FOR SOLVING DIFFERENTIAL EQUATIONS

2020 ◽  
Vol 4 (2) ◽  
pp. 31-33
Author(s):  
Qura Tul Ain ◽  
Muhammad Ali ◽  
Bilqees . ◽  
Muhammad Yousif
Keyword(s):  
Differential Equations ◽  
Galerkin Method ◽  
Ordinary Differential Equations ◽  
Real Life ◽  
High Accuracy ◽  
Rapid Convergence ◽  
Graphical Comparison ◽  
Spectral Galerkin Method ◽  
Pseudo Spectral ◽  
Life Phenomenon

Differential equations have a remarkable potential to exhibit real life phenomenon. Many methods have been developed to solve these differential equations, though only a few stands with time. This paper presents a comparison of Pseudo Spectral Galerkin Method for solving ordinary differential equations with many other global methods. Results shows the high accuracy and rapid convergence of said method. Graphical comparison and error tables have been provided for better understanding of results.

scholarly journals THE FOURIER-ASYMPTOTIC APPROXIMATION AND ITS APPLICATIONS

1998 ◽  
Vol 3 (1) ◽  
pp. 14-24
Author(s):  
Mihails Belovs ◽  
Jānis Smotrovs
Keyword(s):  
Fourier Series ◽  
Differential Equations ◽  
Galerkin Method ◽  
Ordinary Differential Equations ◽  
Asymptotic Approximation ◽  
Fourier Coefficients ◽  
High Accuracy ◽  
Numerical Examples ◽  
Different Types ◽  
Accuracy Of Results

The Fourier ‐ asymptotic approximation can be obtained for different types of Fourier series by replacing the Fourier coefficients with their asymptotic (n → + 8) approximations beginning with some index n. We obtain some generalization of the classical Galerkin method for the solution of boundary and spectral problems of ordinary differential equations. The numerical examples show that the addition of asymptotic correction allows us to obtain a high accuracy of results.

scholarly journals Approximate Solution of Fractional Nonlinear Partial Differential Equations by the Legendre Multiwavelet Galerkin Method

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
M. A. Mohamed ◽  
M. Sh. Torky
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Galerkin Method ◽  
Nonlinear Partial Differential Equations ◽  
Algebraic Equations ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
The Galerkin Method

The Legendre multiwavelet Galerkin method is adopted to give the approximate solution for the nonlinear fractional partial differential equations (NFPDEs). The Legendre multiwavelet properties are presented. The main characteristic of this approach is using these properties together with the Galerkin method to reduce the NFPDEs to the solution of nonlinear system of algebraic equations. We presented the numerical results and a comparison with the exact solution in the cases when we have an exact solution to demonstrate the applicability and efficiency of the method. The fractional derivative is described in the Caputo sense.

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