Variational Quantum Algorithms(VQAs) are a class of algorithm that uses a hybrid approach to solve optimization problems. VQAs use a quantum computer to compute expectation values of circuits that are encoded using classical data and then use a classical optimizer to adjust the parameters of these quantum circuits in order to find the optimal solution. The cost function is defined for any specific problem such that the set of parameters that yields the lowest cost function value corresponds to the solution of that problem. This method shows great potential in Near-term Intermediate Scale Quantum (NISQ) devices. This thesis investigates two approaches of constructing cost functions: the distance minimization approach and the energy minimization approach. Periodic, Dirichlet, and Neumann boundary conditions are studied. We present results for simulation of the heat equation solver, the Poisson equation solver, and the Navier-Stokes equation solver. We also present two ansatz designs, the ZGR-QFT ansatz and the Universal Layered Ansatz, as well as optimization strategies.