HIGHER ORDER OF CONVERGENCE VIA GENERALIZED QUASILINEARIZATION METHOD FOR PARABOLIC INTEGRO-DIFFERENTIAL EQUATIONS

The method of quasilinearization has been generalized so that it is applicable to a wide variety of nonlinear problems. This is known as Generalized Quasulinearization Method (GQM method for short). It has all the advantages of the quasilinearization method such as linear iterates and quadratic convergence. Also it has been developed with a weaker condition than asking for the convexity or concavity of the original qasilineaization method. In this work we will focus on the mathematical models which leads to nonlinear parabolic integro-differential equations. These mathematical models are motivated by population models in biology and the Hodgkin-Huxley model in medicine. We consider the situation when the component functions f (t, x, u) and g(t, x, u) of the orcing function satisfy the following conditions: i) and  exist and they are nondecreasing in u for m > 2; ii)  and  are onesided Lipschitzian with respect the unique solution with the rate of convergence m. The earlier known results on cubic and quadrtic convergence can be obtained as special cases of our current result. A numerical example is presented as an application of our theoretical result. This result is a generalization of GQM method to obtain higher order of convergence for nonlinear parabolic integro-differential equations.