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高校应用数学学报
Applied Mathematics A Journal of Chinese Universities  2014, Vol. 29 Issue (4): 397-411    DOI:
    
A high order schema for the numerical solution of the nonlinear two-dimensional Volterra integral equations
WANG Zi-qiang, CAO Jun-ying
College of Science, Guizhou Minzu University, Guiyang 550025, China
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Abstract  This paper presents a general technique to construct high order schemes for the numerical solutions of the second kind nonlinear two-dimensional Volterra integral equations. This technique is based on the so-called block-by-block approach, which is a common method for the integral equations. In this approach, the classical block-by-block approach is improved in order to avoiding the coupling of the unknown solutions at each block step with an exception at $u(x_1,y),u(x_2,y),u(x,y_1)$ and $u(x,y_2)$, while preserving the good convergence property of the block-by-block schemes. By using this new approach, a high order schema is constructed for the second kind nonlinear two-dimensional Volterra integral equations. The convergence of the schema is rigorously established. It is proved that the numerical solution converges to the exact solution with order $4$.

Key words： nonlinear two-dimensional Volterra integral equations      high order schema      convergence analysis     
Received: 05 February 2014      Published: 08 June 2018
CLC:  O241.82  
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WANG Zi-qiang
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WANG Zi-qiang, CAO Jun-ying. A high order schema for the numerical solution of the nonlinear two-dimensional Volterra integral equations. Applied Mathematics A Journal of Chinese Universities, 2014, 29(4): 397-411.

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http://www.zjujournals.com/amjcua/     OR     http://www.zjujournals.com/amjcua/Y2014/V29/I4/397


非线性二维Volterra积分方程的一个高阶数值格式

对非线性二维Volterra积分方程构造了一个高阶数值格式. block-by-block方法对积分方程来说是一个非常常见的方法, 借助经典block-by-block方法的思想, 构造了一个所谓的修正block-by-block方法. 该方法的优点在于除$u(x_1,y),u(x_2,y),u(x,y_1)$和$u(x,y_2)$外, 其余的未知量不需要耦合求解, 且保存了block-by-block方法好的收敛性. 并对此格式的收敛性进行了严格的分析, 证明了数值解逼近精确解的阶数是4阶.

关键词： 非线性二维Volterra积分方程,  高阶格式,  收敛性分析 
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