The theme of the thesis can be devided to two major aspects.The first is a treatment of non- classical variational approach for every linear operator, by studying Magri's method and Riess and Haug method. Also, a simpler version of Magri's method is discussed and it's validity is proved when the linear operator takes a special form.Many examples with non symmetric linear operators such as differential equations, integral equations; and integro- differential equations, and systems of such equations are solved numerically by the use of the novel non-classical variational technique. The second aspect of the thesis deals with finding the solution of moving boundary value problems, especially, the problem of oxygen diffusion in biological tissues. This problem is solved using the non- classical variational tools . For each considered case a computer program is written and implemented; the results are given in tabular, and graphical forms with a comparison with the exact solution or with solution found by other means, with very reasonable agreements.