Ram N. Mohapatra

The explicit series solution of SIR and SIS epidemic models

In this paper the SIR and SIS epidemic models in biology are solved by means of an analytic technique for nonlinear problems, namely the homotopy analysis method (HAM). Both of the SIR and SIS models are described by coupled nonlinear differential equations. A one-parameter family of explicit series solutions are obtained for both models. This parameter has no physical meaning but provides us with a simple way to ensure convergent series solutions to the epidemic models. Our analytic results agree well with the numerical ones. This analytic approach is general and can be applied to get convergent series solutions of some other coupled nonlinear differential equations in biology.

Existence and nonuniqueness of solutions of a singular nonlinear boundary-layer problem

Abstract Sufficient conditions for existence and nonuniqueness of positive solutions of the singular boundary value problem g ( x ) g ″( x ) + h ( x ) = 0, − k ⩽ x k > 0, g ′(− k ) = C , g (1) = 0 are obtained. Also, it is proved that the solutions with g(−k) > −Ck ( for C and g(−k) > ( k 2 ) √ −2h(−k) ( for C > 0) are unique. Furthermore, it is shown numerically that for h ( x ) = x there are exactly two Solutions for the problem.

Extension of the method of quasilinearization and rapid convergence

An extension of the method of quasilinearization has been applied to first-order nonlinear initial-value problems (IVP for short). It has been shown that there exist monotone sequences which converge rapidly to the unique solution of IVP.

On solutions of some singular, non-Linear differential equations arising in boundary layer theory

Abstract Solutions for a class of singular, non-linear, second-order differential equations arising in boundary layer theory with suction/injection, when Crocco variables are employed, are obtained. Existence, uniqueness, and analyticity results are established for boundary conditions corresponding to flow of a uniform stream past a semi-infinite flat plate (classical problem of Blasius) and for the flow behind weak expansion. Since the standardization technique (in Refs. [8, 9, 11]) does not work, a new technique is developed and used in proving existence and uniqueness theorems. Furthermore, the analytical solutions are compared with the numerical ones.

Subordinations for analytic functions defined by the Dziok-Srivastava linear operator

In the present investigation, we obtain certain sufficient conditions for a normalized analytic function f(z) defined by the Dziok–Srivastava linear operator H l ½a1� to satisfy the certain subordination. Our results extend corresponding previously known results on starlikeness, convexity, and close to convexity.

Nonsmooth ρ − (η, θ)-invexity in multiobjective programming problems

In this paper we extend Reiland’s results for a nonlinear (single objective) optimization problem involving nonsmooth Lipschitz functions to a nonlinear multiobjective optimization problem (MP) for ρ − (η, θ)-invex functions. The generalized form of the Kuhn–Tucker optimality theorem and the duality results are established for (MP).

A certain family of mixed summation-integral type operators

In the present paper, we study the mixed summation integral type operators having different weight functions, we obtain the rate of point wise convergence, an asymptotic formula of Voronovskaja type and some local direct results in terms of modulus of smoothness and modulus of continuity in ordinary and simultaneous approximation.

Constrained quadratic correlation filters for target detection

A method for designing and implementing quadratic correlation filters (QCFs) for shift-invariant target detection in imagery is presented. The QCFs are quadratic classifiers that operate directly on the image data without feature extraction or segmentation. In this sense the QCFs retain the main advantages of conventional linear correlation filters while offering significant improvements in other respects. Not only is more processing required for detection of peaks in the outputs of multiple linear filters but choosing the most suitable among them is an error-prone task. All channels in a QCF work together to optimize the same performance metric and to produce a combined output that leads to considerable simplification of the postprocessing scheme. The QCFs that are developed involve hard constraints on the output of the filter. Inasmuch as this design methodology is indicative of the synthetic discriminant function (SDF) approach for linear filters, the filters that we develop here are referred to as quadratic SDFs (QSDFs). Two methods for designing QSDFs are presented, an efficient architecture for achieving them is discussed, and results from the Moving and Stationary Target Acquisition and Recognition synthetic aperture radar data set are presented.

On the divergence of Lagrange interpolation with equidistant nodes

This paper is concerned with the optimal rate of divergence of Lagrange interpolation of f(x) = |x| at equidistant nodes

Approximation solvability of a class of A-monotone implicit variational inclusion problems in semi-inner product spaces

Abstract This paper deals with the existence of solutions for a class of nonlinear implicit variational inclusion problems in semi-inner product spaces. We construct an iterative algorithm for approximating the solution for the class of implicit variational inclusions problems involving A -monotone and H -monotone operators by using the generalized resolvent operator technique.