In this dissertation, a new method based on the method of fundamental solutions has been proposed for solution of partial differential equations (PDEs) with constant coefficients. The first set of the results has been presented for the Helmholtz equation. In this method, the approximate solution of the homogeneous equation is expressed as a series, using exponential functions as the fundamental solutions. Constant coefficients of this series are evaluated through an especial discrete transformation. However, the results accuracy is highly dependent on the number, amplitude and fluctuations of the implemented fundamental solutions. Hence, finding a suitable pattern for choosing the associated parameters of the fundamental solutions plays an important role in reducing the computational error and reaching appropriate accuracy in final results. Two different methods are also proposed for approximating the particular solution of the Helmholtz equation; i.e. the Fourier series solution and a new method called “Exponential series” based on the discrete transformation. Comparison of these two methods reveals significant supremacy of the latter over the former in almost all problems regarding the computational costs. After developing an appropriate algorithm for solving the Helmholtz equation, this algorithm has been extended to other important PDEs in solid mechanics including the equations of elasticity, elastic wave, static thin/thick plates and forced transverse vibration of plates. The solution of advection-diffusion-absorption problems has also been addressed. Several benchmark examples have been solved in each category of problems. It has been concluded that this method is capable of solving miscellaneous problems, with excellent precision and accuracy, on complicated domain shapes with various types of boundary conditions. The method has also been employed in an element based manner. Element implementation in discretizing the domain obviates some limitations of the method of fundamental solutions and results in a considerable increase in accuracy in certain cases.
