The present work considers the approximation of solutions of a type of fractional order Volterra-Fredholm integro-differential equations, where the fractional derivative is introduced in Caputo sense. In addition, we also present several applications of the fractional order differential equations and integral equations. Here, we provide a sufficient condition for existence and uniqueness of the solution and also obtain an a priori bound of the solution of the present problem. Then, we discuss about the higher order model equation which can be written as a system of equations whose orders are less than or equal to one. Next, we present an approximation of the solution of this problem by means of a perturbation approach based on homotopy analysis. Also, we discuss the convergence analysis of the method. It is observed through different examples that the adopted strategy is a very effective one for good approximation of the solution, even for higher order problems. It is shown that the approximate solutions converge to the exact solution, even for higher order fractional differential equations. In addition, we show that the present method is highly effective compared to the existed method and produces less error.

This paper studies a higher order numerical method for the singularly perturbed parabolic convection-diffusion problems where the diffusion term is multiplied by a small perturbation parameter. In general, the solutions of these type of problems have a boundary layer. Here, we generate a spatial adaptive mesh based on the equidistribution of a positive monitor function. Implicit Euler method is used to discretize the time variable and an upwind scheme is considered in space direction. A higher order convergent solution with respect to space and time is obtained using the postprocessing based extrapolation approach. It is observed that the convergence is independent of perturbation parameter. This technique enhances the order of accuracy from first order uniform convergence to second order uniform convergence in space as well as in time. Comparative study with the existed meshes show the highly effective behavior of the present method.