A collocation method based on reproducing kernel for a modified anomalous subdiffusion equation (Q2874178)

A modified anomalous subdiffusion partial differential equation containing fractional-time derivatives acting on a diffusion operator is considered. Applications of this equation include problems in continuous-time random-walk models in financial markets. A numerical technique to solve this kind of problems is proposed and analyzed. It is based on using three reproducing kernel Hilbert spaces associated with different parts of the equation. With these reproducing kernels, it is shown that the original differential equation is equivalent to the equation \(L v = g_1\), where \(v(x,t)\) is the unknown function, \(g_1\) is a given continuous function and \(L\) is a bounded linear operator defined in one of the reproducing kernels previously introduced. This equivalence allows the authors to introduce the concept of an \(\varepsilon\)-approximate solution as that \(v\) such that \(\| L v - g_1 \| < \varepsilon\), which is then constructively determined by obtaining the coefficients of \(v\) in an optimal way by solving a system of linear equations. The technique is illustrated with a pair of numerical examples.