Application of Müntz Orthogonal Functions on the Solution of the Fractional Bagley–Torvik Equation Using Collocation Method with Error Stimate

IF 1 4区 物理与天体物理 Q3 PHYSICS, MATHEMATICAL
S. Akhlaghi, M. Tavassoli Kajani, M. Allame
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引用次数: 0

Abstract

This paper uses Müntz orthogonal functions to numerically solve the fractional Bagley–Torvik equation with initial and boundary conditions. Müntz orthogonal functions are defined on the interval 0 , 1 and have simple and distinct real roots on this interval. For the function f ∈ L 2 0 , 1 , we obtain the best unique approximation using Müntz orthogonal functions. We obtain the Riemann–Liouville fractional integral operator for Müntz orthogonal functions so that we can reduce the complexity of calculations and increase the speed of solving the problem, which can be seen in the process of running the Maple program. To solve the fractional Bagley–Torvik equation with initial and boundary conditions, we use Müntz orthogonal functions and consider simple and distinct real roots of Müntz orthogonal functions as collocation points. By using the Riemann–Liouville fractional integral operator that we define for the Müntz orthogonal functions, the process of numerically solving the fractional Bagley–Torvik equation that is solved using Müntz orthogonal functions is reduced, and finally, we reach a system of algebraic equations. By solving algebraic equations and obtaining the vector of unknowns, the fractional Bagley–Torvik equation is solved using Müntz orthogonal functions, and the error value of the method can be calculated. The low error value of this numerical solution method shows the high accuracy of this method. With the help of the Müntz functions, we obtain the error bound for the approximation of the function. We have obtained the error bounds for the numerical method using which we solved the fractional Bagley–Torvik equation with initial and boundary conditions. Finally, we have given a numerical example to show the accuracy of the solution of the method presented in this paper. The results of solving this example using Müntz orthogonal functions and comparing the results with other methods that have been used the solve this example show the higher accuracy of the method proposed in this paper.
m ntz正交函数在带误差估计的配点法解分数阶Bagley-Torvik方程中的应用
本文利用m ntz正交函数对具有初始条件和边界条件的分数阶Bagley-Torvik方程进行了数值求解。在区间(0,1)上定义了m ntz正交函数,并在此区间上具有简单且不同的实根。对于函数f∈l2,1,我们利用m ntz正交函数得到了最佳唯一逼近。我们得到了m ntz正交函数的Riemann-Liouville分数积分算子,从而降低了计算的复杂性,提高了求解问题的速度,这在Maple程序的运行过程中可以看出。为了求解具有初始条件和边界条件的分数阶Bagley-Torvik方程,我们使用m ntz正交函数,并考虑m ntz正交函数的简单且不同的实根作为配点。利用为m ntz正交函数定义的Riemann-Liouville分数阶积分算子,简化了用m ntz正交函数求解分数阶Bagley-Torvik方程的数值求解过程,最终得到了一个代数方程组。通过求解代数方程,得到未知数向量,利用m ntz正交函数求解分数阶Bagley-Torvik方程,并计算出该方法的误差值。该数值解的误差值较小,说明该方法具有较高的精度。借助m ntz函数,我们得到了函数逼近的误差界。得到了具有初始条件和边界条件的分数阶Bagley-Torvik方程数值求解方法的误差范围。最后给出了一个数值算例,说明了本文方法求解的准确性。利用m ntz正交函数求解该算例的结果,并与求解该算例的其他方法的结果进行了比较,表明本文方法具有较高的精度。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Advances in Mathematical Physics
Advances in Mathematical Physics 数学-应用数学
CiteScore
2.40
自引率
8.30%
发文量
151
审稿时长
>12 weeks
期刊介绍: Advances in Mathematical Physics publishes papers that seek to understand mathematical basis of physical phenomena, and solve problems in physics via mathematical approaches. The journal welcomes submissions from mathematical physicists, theoretical physicists, and mathematicians alike. As well as original research, Advances in Mathematical Physics also publishes focused review articles that examine the state of the art, identify emerging trends, and suggest future directions for developing fields.
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