This work is concerned with the development of a scheme to obtain boundary conditions adapted representations of derivatives (differential operators) in the orthonormal wavelet bases of Daubechies family in \(\Omega = [0, \ 1] \subset {\mathbb{R}}\). The scheme is employed to find approximate solution of the (1+1)-D advection-diffusion-transport equation arising in the flow of contaminant fluid (water) through a porous medium (ground). Representation of derivatives adopting Dirichlet’s boundary conditions have been derived first. These representations are then used to reduce the aforesaid advection-diffusion-transport equation to a system of nonhomogeneous first order coupled ordinary differential equations. It is observed that representations of the derivatives derived here shift the non-homogeneous terms in the boundary conditions to non-homogeneous terms of the reduced ordinary differential equations. The resulting ordinary differential equations can be solved globally in time, in general, which helps to prevent the necessary analysis of time discretization such as estimate of step lengths, error etc. In case of convection dominated problems, the system of ordinary differential equations can be solved numerically with the aid of any efficient ODE solver. An estimate of \(L^2\)-error in wavelet-Galerkin approximation of the unknown solution has been presented. A number of examples is given for numerical illustration. It is found that the scheme is efficient and user friendly.

这项工作涉及开发一种方案，以在\(\Omega = [0, \ 1] \subset {\mathbb{R} }\). 该方案用于找到在污染物流体（水）通过多孔介质（地面）流动时出现的 (1+1)-D 对流-扩散-输运方程的近似解。已经首先导出了采用狄利克雷边界条件的导数的表示。然后使用这些表示将上述对流-扩散-输运方程简化为非齐次一阶耦合常微分方程组。观察到，此处导出的导数的表示将边界条件中的非齐次项转移到简化的常微分方程的非齐次项。由此产生的常微分方程可以在时间上全局求解，一般来说，这有助于防止对时间离散化进行必要的分析，例如估计步长、误差等。在对流主导问题的情况下，可以借助任何有效的 ODE 求解器对常微分方程组进行数值求解。估计\(L^2\) - 未知解的小波伽辽金近似误差已经被提出。给出了许多示例用于数值说明。发现该方案高效且用户友好。