An accurate RBF–based meshless technique for the inverse multi-term time-fractional integro-differential equation

Approximation of three‐dimensional nonlinear wave equations by fundamental solutions and weighted residuals process

“…For a fixed positive number $$$ {h}_t={t}_{l+1}-{t}_l $$$, the grid points in the time interval $$$ \left[0,T\right] $$$ are labeled $$$ {t}_l=l{h}_t $$$. It is well known that the approximation of the first‐ and second‐order derivatives is as follows [26] $$$$ {\displaystyle \begin{array}{ll}\frac{\partial^2{C}^{l+1}}{\partial {t}^2}& \approx \frac{C^{l+1}-2{C}^l+{C}^{l-1}}{h_t^2},\\ {}\frac{\partial {C}^{l+1}}{\partial t}& \approx \frac{C^{l+1}-{C}^{l-1}}{2{h}_t},\end{array}} $$$$ where $$$ {C}^{l+1}=C\left(\boldsymbol{x},{t}_{l+1}\right) $$$. Also we employ the following approximations by using the CN technique: …”