Representation of exact solution for the time-fractional telegraph equation in the reproducing kernel space

The reproducing kernel theorem is used to solve the time-fractional telegraph equation with Robin boundary value conditions. The time-fractional derivative is considered in the Caputo sense. We discuss and derive the exact solution in the form of series with easily computable terms in the reproducing kernel space.

Research highlights
► Reproducing kernel space W(Ω)W(Ω) is constructed. Converting the fractional equation to an operator equation Lu(x,t)=F(x,t)Lu(x,t)=F(x,t) in (5), where L:W(Ω)→L2(Ω)L:W(Ω)→L2(Ω).
► Constructing a family of orthogonal basis {ψi¯(x,t)}i=1∞ in reproducing kernel space.
► The solution of the operator equation is expanded by series according to the orthogonal basis, u(x,t)=∑i=1∞∑k=1iβikF(xk,tk)ψi¯(x,t).
► Numerical experiment.