We derive a method of fundamental solutions (MFS) for the numerical solution of an ill-posed lateral Cauchy problem for the hyperbolic wave equation in bounded planar annular domains. The Laguerre transform is applied to reduce the time-dependent lateral Cauchy problem to a sequence of elliptic Cauchy problems with a known set of fundamental solutions termed a fundamental sequence. The solution of the elliptic problems is approximated by linear combinations of the elements in the fundamental sequence. Source points are placed outside of the solution domain, and by collocating on the boundary of the solution domain itself a sequence of linear equations is obtained for finding the coefficients in the MFS approximation. It is shown that the fundamental solutions used constitute a linearly independent and dense set on the boundary of the solution domain with respect to the L₂-norm. Tikhonov regularization is applied to get a stable solution to the obtained systems of linear equations in combination with the L-curve rule for selecting the regularization parameter. Numerical results confirm the efficiency and applicability of the proposed strategy for the considered lateral Cauchy problem both in the case of exact and noisy data. Adjusting the coefficients in the sequence of elliptic equations, the similar strategy and results apply also to the parabolic lateral Cauchy problem as verified by an included numerical example. It is also shown that by adjusting the coefficients further the method of Rothe can be applied as an alternative to the Laguerre transformation in time.

我们导出了一种有界平面环域中双曲波方程不适定横向柯西问题数值解的基本解（MFS）方法。使用Laguerre变换将时间相关的横向柯西问题简化为具有一组称为基本序列的已知基本解的椭圆柯西问题序列。椭圆问题的解决方案通过基本序列中元素的线性组合来近似。源点位于解域的外部，并且通过并置在解域本身的边界上，获得了一系列线性方程组，用于查找MFS近似中的系数。L _2-范数。结合L曲线规则选择正则化参数，应用Tikhonov正则化对所获得的线性方程组进行稳定求解。数值结果证实了在精确和嘈杂数据情况下所提出的策略对于所考虑的侧向柯西问题的有效性和适用性。调整椭圆方程序列中的系数，类似的策略和结果也适用于抛物线型侧向柯西问题，这一点已通过一个包含的数值示例进行了验证。还表明，通过进一步调整系数，可以将Rothe方法用作Laguerre变换的替代方法。