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In this paper, we introduce a new classTs, a k [A, B, a, b] of analytic functions by using a newly defined convolution Operator. This class contains many known classes of analytic and univalent functions as special cases. We derived some interesting results including inclusion relationships, a radius problem and sharp coefficient bound for this class.

In this paper,  a  one-dimensional modified Burgers'  equation is considered for different  Reynolds numbers. For very high Reynolds numbers,  the solution possesses a multiscale character in some part of the independent domain and thus can be classified as a  singularly perturbed problem. A numerical scheme that uses a fitted Operator finite difference scheme to solve the spatial derivatives and the implicit Euler scheme for the time derivative is proposed to solve the modified  Burgers'  equation via Rothe's method. It is important to note that the proposed fitted Operator finite difference scheme is based on the midpoint upwind scheme. The stability of the scheme is established and the error associated with each discretisation is estimated. Numerical simulations are carried out to validate the theoretical findings.

A second order finite difference scheme is constructed to solve a singularly perturbed degenerate parabolic convection diffusion problem via Rothe's method. The solution of the problem exhibits a boundary layer on the left side of the spatial domain.  By means of the Crank Nicolson finite difference scheme, the time derivative is discretised to obtain a set of semi-discrete boundary value problems. Using a fitted Operator finite difference scheme based on the midpoint downwind scheme,  the system of boundary value problems are discretized and analysed for convergence. Second order accuracy is established for each discretisation process. Numerical simulations are carried out to validate the theoretical error estimate.

In this literature, we study the existence and stability of the solution of the boundary value problem of fractional di, erential equations with , p-Laplacian Operator. Our problem is based on Caputo fractional derivative of orders , ,ϵ, , where k 􀀀, 1 < , ,ϵ,,k, and k ,3. By using the Schauder , xed point theory and properties of the Green function, some conditions are established which show the criterion of the existence and non-existence solution for the proposed problem. We also investigate some adequate conditions for the Hyers-Ulam stability of the solution. Illustrated examples are given as an application of our result.