Dissertation (MSc Mathematics)Present work is concerned with solved a coefficient inverse problem of one-dimensional parabolic equation by a higher-order compact finite difference method and we used this a fourth order efficient numerical method to calculate the function u (x; t) and the unknown coefficient a (t) in a parabolic partial differential equation. Also discussed the accuracy and efficiency of the fourth order finite difference formula compare with other finite difference methods such as FTCS explicit scheme, Crank-Necolson algorithm and Back ward time central space scheme. Results show that an excellent estimation on the unknown functions of the inverse problem can be obtained and the fourth order method developed in this work is well-balanced in stability, efficiency and accuracy.