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.

In this paper, we determine the upper and lower bounds for the norm of lower triangular matrix operators on Cesà, ro weighted (p, v)−, fractional difference sequence spaces of modulus functions. We consider the matrix operators acting between ℓ, p(w) and Cp(v, ω, , Δ, (η, , ℓ, ), F) and identify their bounds and vice-versa. We also investigate the same characteristics for Nö, rlund and weighted mean matrix operators.

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.

The Schwarz inequality and Jensen’s one are fundamental in a Hilbert space. Regarding a sesquilinear map B(X, Y ) = Y*X as an operatorvalued inner product, we discuss operator versions for the above inequalities and give simple conditions that the equalities hold.

In this investigation, using Opoola differential operator ($D^{m}(\mu,\beta,t)f(z)$), a new integral operator: $I_{t,\beta,\mu}^{m,\sigma}(f_{1},...,f_{n})(z): A^{n}\rightarrow A$  is defined in the unit disk, $U=\left\lbrace z\in C:\left|z\right|<1\right\rbrace$; and we investigated the Univalence conditions of this generalized operator. Finally, a number of corollaries and remarks which show the extension of our results are presented.