In this paper, we propose an efficient non-linear task workload prediction mechanism incorporated with a fair scheduling algorithm for task allocation and resource management in Grid computing. Workload prediction is accomplished in a Grid middleware approach using a non-linear model expressed as a series of finite known functional components using concepts of functional analysis. The coefficient of functional components are obtained using a training set of appropriate samples, the pairs of which are estimated based on a runtime estimation model relied on a least squares approximation scheme. The advantages of the proposed non-linear task workload prediction scheme is that (i) it is not constrained by analysis of source code (analytical methods), which is practically impossible to be implemented in complicated real-life applications or (ii) it does not exploit the variations of the workload statistics as the statistical approaches does. The predicted task workload is then exploited by a novel scheduling algorithm, enabling a fair Quality of Service oriented resource management so that some tasks are not favored against others. The algorithm is based on estimating the adjusted fair completion times of the tasks for task order selection and on an earliest completion time strategy for the grid resource assignment. Experimental results and comparisons with traditional scheduling approaches as implemented in the framework of European Union funded research projects GRIA and GRIDLAB grid infrastructures have revealed the outperformance of the proposed method.