In this manuscript, we aim to study the semi-analytical and the numerical solution of a semilinear time-fractional diffusion equation where the time-fractional term includes the combination of tempered fractional derivative and k-Caputo fractional derivative with a parameter \(k \ge 1\) . The application of the new integral transform, namely Elzaki transform of the tempered k-Caputo fractional derivative is shown here and thereafter the semi-analytical solution is obtained by using the Elzaki decomposition method. The model problem is linearized using Newton’s quasilinearization method, and then the quasilinearized problem is discretized by the difference scheme namely tempered \(_kL2\) - \(1_\sigma \) method. Stability and convergence analysis of the proposed scheme have been discussed in the \(L_2\) -norm using the energy method. In support of the theoretical results, numerical example has been incorporated.
