一类半线性椭圆方程柯西问题的正则化方法

One Regularization Method for a Cauchy Problem of Semilinear Elliptic Equation

构造并利用一种广义分数Tikhonov正则化方法研究一类半线性椭圆方程柯西问题.基于所构造的正则化解满足一个非线性积分方程,首先证明正则化解的存在唯一性和稳定性;继而在对精确解的先验假设下给出并证明正则化方法的收敛性;最后设计一种迭代算法计算正则化解,并通过相应的计算结果验证了所提方法的稳定可行性.

In this paper,we construct and use a generalized fractional Tikhonov regularization method to study a Cauchy problem for semi-linear elliptic equation.Based on one nonlinear integral equation that the constructed regularization solution satisfies,we firstly prove the existence,uniqueness and stability for it.And then we give and prove the convergence for regularized method under an a-priori assumption on the exact solution.Ultimately,the regularized solution is calculated by designing an iteration algorithm,and we verify the stability and feasibility for proposed method by the corresponding computational results.