ABDULAZIZ ABDULLAH OTHMAN AL-SHUAIBI2022-05-182022-05-185024https://drepo.sdl.edu.sa/handle/20.500.14154/2709The objective of this work is to develop new methods for the numerical inversion of the Laplace transform. The 1st method is an approximation by Legendre polynomials and an estimation of the error bounds via some optimal parameters shall be derived. The 2nd method overcomes the ill-posedness by using the Tikhonov regularization. The numerical treatment is based on convolution of Hermite polynomials approximations. The spectral analysis provides a spectral representation leading to a real inversion formula, and by using pseudo-differential operators techniques, we shall obtain a direct approximation method.enThesis