Abstarct: In the present study the flow and heat transfer of viscoelastic fluids passing through straight rectangular ducts under constant heat flux on the walls and in the fully developed hydrodynamic and thermal region have been investigated analytically. The nonlinear constitutive equation used to model the studied viscoelastic fluid is the FENE-P model which is one of the popular models for modeling dilute polymer solutions. In this study the Ritz method is used to solve the flow field and heat transfer of viscoelastic fluid, which is subset of calculus of variations methods for solving partial differential equations. Comparison of the results obtained for the flow field with previous studies confirms the high efficiency and accuracy of the Ritz method. Also, using the fourth-order Ritz method is sufficient for the accuracy of four decimal places. After solving the FENE-P viscoelastic fluid flow field, the effect of dimensionless parameters such as Deborah number, extensibility parameter and channel cross-section aspect ratio have been investigated on velocity distribution. The results show that with increasing the Deborah number and decreasing the extensibility parameter, the maximum velocity in the center of channel increases due to the intensification of shear thinning behavior of viscoelastic fluid. Also, increasing the cross-section aspect ratio will reduce the maximum velocity. By dividing the obtained velocity distribution by its average velocity, it was found that both an increase of the Deborah number or a decrease of the extensibility parameter lead to flatter velocity profiles in the core region as a consequence of enhanced shear thinning in viscosity. In solving the energy equation by considering viscous dissipation, in addition to the mentioned dimensionless parameters, the effect of Brinkman number on heat transfer of FENE-P viscoelastic fluid has been analyzed. The results show that with increasing Deborah number and decreasing the extensibility parameter, the amount of viscous dissipation around the walls increases. For positive Brinkman numbers, where heat is transferred from the walls to the fluid, the mean temperature of the fluid decreases as the Deborah number increases and the extensibility parameter decreases, which leads to reduction in Nusselt numbers. But for negative Brinkman numbers, where heat is transferred from the fluid to the walls, this trend will be reversed. The results reveal that by increasing the positive and negative Brinkman numbers in the specific Deborah number and extensibility parameter, the Nusselt numbers will decrease and increase, respectively.