Abstract:
Indian Cancer Research Centre (ICRC) bacillus strain (C-44), a candidate vaccine against leprosy is cultured in vitro in Dubos medium enriched with amino acids and human serum. The study was conducted to find out whether ICRC bacilli obtained from these cultures are coated with anti mycobacterial antibody. Anti-ICRC antibody raised by intradermal inoculation of sonicated ICRC bacilli in rabbits reacted with both human immunoglobulin G (IgG) and antigens of ICRC. Further, ICRC bacilli could also be fluoresced directly with FITC labelled anti human IgG. Positive fluorescence of ICRC could be abolished by digestion of human IgG with trypsin and carbon tetrachloride (CCL4). It is concluded that ICRC bacilli present in the vaccine are coated with human IgG.

Abstract:
we propose a new high-order approximation for the solution of two-space-dimensional quasilinear hyperbolic partial differential equation of the form =(,,,)

Abstract:
In this piece of work using only three grid points, we propose two sets of numerical methods in a coupled manner for the solution of fourth-order ordinary differential equation , , subject to boundary conditions , , , and , where , , , and are real constants. We do not require to discretize the boundary conditions. The derivative of the solution is obtained as a byproduct of the discretization procedure. We use block iterative method and tridiagonal solver to obtain the solution in both cases. Convergence analysis is discussed and numerical results are provided to show the accuracy and usefulness of the proposed methods. 1. Introduction Consider the fourth-order boundary value problem subject to the prescribed natural boundary conditions or equivalently, for , subject to the natural boundary conditions where , , , and are real constants and . Fourth-order differential equations occur in a number of areas of applied mathematics, such as in beam theory, viscoelastic and inelastic flows, and electric circuits. Some of them describe certain phenomena related to the theory of elastic stability. A classical fourth-order equation arising in the beam-column theory is the following (see Timoshenko [1]): where is the lateral deflection, is the intensity of a distributed lateral load, is the axial compressive force applied to the beam, and represents the flexural rigidity in the plane of bending. Various generalizations of the equation describing the deformation of an elastic beam with different types of two-point boundary conditions have been extensively studied via a broad range of methods. The existence and uniqueness of solutions of boundary value problems are discussed in the papers and book of Agarwal and Krishnamoorthy, Agarwal and Akrivis (see [2–5]). Several authors have investigated solving fourth-order boundary value problem by some numerical techniques, which include the cubic spline method, Ritz method, finite difference method, multiderivative methods, and finite element methods (see [6–16]). In the 1980s, Usmani et al. (see [17–19]) worked on finite difference methods for solving and finite difference methods for computing eigenvalues of fourth-order linear boundary value problem. In 1984, Twizell and Tirmizi (see [20]) developed multi-derivative methods for linear fourth-order boundary value problems. In 1984, Agarwal and Chow (see [21]) developed iterative methods for a fourth-order boundary value problem. In 1991, O’Regan (see [13]) worked on the solvability of some fourth-(and higher) order singular boundary value problems. In 1994, Cabada (see

Abstract:
We discuss a new single sweep alternating group explicit iteration method, along with a third-order numerical method based on off-step discretization on a variable mesh to solve the nonlinear ordinary differential equation subject to given natural boundary conditions. Using the proposed method, we have solved Burgers’ equation both in singular and nonsingular cases, which is the main attraction of our work. The convergence of the proposed method is discussed in detail. We compared the results of the proposed iteration method with the results of the corresponding double sweep alternating group explicit iteration methods to demonstrate computationally the efficiency of the proposed method. 1. Introduction Consider the general nonlinear ordinary differential equation subject to essential boundary conditions where are finite constants. We assume that for (i) is continuous,(ii) and exist and are continuous,(iii) and for some positive constant . These conditions ensure that the boundary value problem (1) and (2) possesses a unique solution (see Keller [1]). With the advent of parallel computers, scientists are focusing on developing finite difference methods with the property of parallelism. Working on this, in the early 1980s, Evans [2, 3] introduced the Group Explicit methods for large linear system of equations. Further he discussed the Alternating Group Explicit (AGE) method to solve periodic parabolic equations in a coupled manner. Mohanty and Evans applied AGE method along with various high order methods [4, 5] for the solution of two-point boundary value problems. Later, Sukon and Evans [6] introduced a Two-parameter Alternating Group Explicit (TAGE) method for the two-point boundary value problem with a lower order accuracy scheme. In 2003 Mohanty et al. [7] discussed the application of TAGE method for nonlinear singular two point boundary value problems using a fourth-order difference scheme. In 1990, Evans introduced the Coupled Alternating Group Explicit method [8] and applied it to periodic parabolic equations. Many scientists are applying these parallel algorithms to solve ordinary and partial differential equations [9–11]. Recently, Mohanty [12] has proposed a high order variable mesh method for nonlinear two-point boundary value problem. Mohanty and Khosla [13, 14] also devised a new third-order accurate arithmetic average variable mesh method for the solution of the boundary value problem (1) and (2), using three grid points, which is applicable to both singular and nonsingular problems. No special technique is required to handle singular

Abstract:
Al-12.5 wt% Si alloy powder with 15 wt% SiCp was mechanically alloyed (MA) using attrition mill in purified nitrogen atmosphere. The MA processed powder was found to have nano grain size and uniform distribution of SiCp in the AlSi matrix. This MA processed powder was used for atmospheric plasma spraying (APS) for varying distances and currents densities. The coatings obtained were studied by image analyzer, SEM and XRD. Microhardness and wear rate of the coatings were evaluated using Vickers indenter and pin on disk type tribometer, respectively. Adhesion strength of the coatings was measured by interfacial indentation test. The results showed that these coatings have uniform distribution of reinforced SiC particles in the nano crystalline matrix, low porosity (1% - 2%), low wear rates and improved adhe-sion strength. It was also observed that by increasing current density of APS, the adhesive strength increased.

Abstract:
We propose a three-level implicit nine point compact finite difference formulation of order two in time and four in space direction, based on nonpolynomial spline in compression approximation in -direction and finite difference approximation in -direction for the numerical solution of one-dimensional wave equation in polar coordinates. We describe the mathematical formulation procedure in detail and also discussed the stability of the method. Numerical results are provided to justify the usefulness of the proposed method. 1. Introduction We consider the one-dimensional wave equation in polar forms: with the following initial conditions: and the following boundary conditions: where and . We assume that the conditions (2) and (3) are given with sufficient smoothness to maintain the order of accuracy in the numerical method under consideration. The study of wave equation in polar form is of keen interest in the fields like acoustics, electromagnetic, fluid dynamics, mathematical physics, and so forth. Efforts are being made to develop efficient and high accuracy finite difference methods for such types of PDEs. During the last three decades, there has been much effort to develop stable numerical methods based on spline approximations for the solution of time-dependent partial differential equations. But so far in the literature, very limited spline methods are there for the wave equation in polar coordinates. In 1968-69, Bickley [1] and Fyfe [2] studied boundary value problems using cubic splines. In 1973, Papamichael and Whiteman [3], and the next year, Fleck [4] and Raggett and Wilson [5] have used a cubic spline technique of lower order accuracy to solve one-dimensional heat conduction equation and wave equation, respectively. Then, Jain et al. [6–9] have derived cubic spline solution for the differential equations including fourth order cubic spline method for solving the nonlinear two point boundary value problems with significant first derivative terms. Recently, Kadalbajoo et al. [10, 11] and Khan et al. [12, 13] have studied parametric cubic spline technique for solving two point boundary value problems. In recent years, Rashidinia et al. [14], Ding and Zhang [15], and Mohanty et al. [16–21] have discussed spline and high order finite difference methods for the solution of hyperbolic equations. In this present paper, we follow the idea of Jain and Aziz [7] by using nonpolynomial spline in compression approximation to develop order four method in space direction for the wave equation in polar co-ordinates. We have shown that our method is in general

Abstract:
Numerical method based on three geometric stencils has been proposed for the numerical solution of nonlinear singular fourth-order ordinary differential equations. The method can be easily extended to the sixth-order differential equations. Convergence analysis proves the third-order convergence of the proposed scheme. The resulting difference equations lead to block tridiagonal matrices and can be easily solved using block Gauss-Seidel algorithm. The computational results are provided to justify the usefulness and reliability of the proposed method. 1. Introduction Consider the fourth-order boundary value problem: subject to the necessary boundary conditions: where and , , , and are real constants and . Or equivalently subject to the natural boundary conditions: Fourth-order differential equations occur in various areas of mathematics such as viscoelastic and inelastic flows, beam theory, Lifshitz point in phase transition physics (e.g., nematic liquid crystal, crystals, and ferroelectric crystals) [1], the rolls in a Rayleigh-Benard convection cell (two parallel plates of different temperature with a liquid in between) [2], spontaneous pattern formation in second-order materials (e.g., polymeric fibres) [3], the waves on a suspension bridge [4, 5], geological folding of rock layers [6], buckling of a strut on a nonlinear elastic foundation [7], traveling water waves in a shallow channel [8], pulse propagation in optical fibers [9], system of two reaction diffusion equation [10], and so forth. The existence and uniqueness of the solution for the fourth and higher-order boundary value problems have been discussed in [11–14]. In the recent past, the numerical solution of fourth-order differential equations has been developed using multiderivative, finite element method, Ritz method, spline collocation, and finite difference method [15–18]. The determination of eigen values of self adjoint fourth-order differential equations was developed in [19] using finite difference scheme. The motivation of variable mesh technique for differential equations arises from the theory of electrochemical reaction-convection-diffusion problems in one-dimensional space geometry [20]. The geometric mesh method for self-adjoint singular perturbation problems using finite difference approximations was discussed in [21]. The use of geometric mesh in the context of boundary value problems was studied extensively in [22–24]. In this paper, we derive a geometric mesh finite difference method for the solution of fourth- and sixth-order differential boundary value problems with order

Abstract:
An efficient algorithm for the numerical solution of higher (even) orders two-point nonlinear boundary value problems has been developed. The method is third order accurate and applicable to both singular and nonsingular cases. We have used cubic spline polynomial basis and geometric mesh finite difference technique for the generation of this new scheme. The irreducibility and monotone property of the iteration matrix have been established and the convergence analysis of the proposed method has been discussed. Some numerical experiments have been carried out to demonstrate the computational efficiency in terms of convergence order, maximum absolute errors, and root mean square errors. The numerical results justify the reliability and efficiency of the method in terms of both order and accuracy. 1. Introduction Consider the following nonlinear two-point boundary value problems of order : where , , and , are finite real constants and . The higher order two-point boundary value problems play an important role in various areas of mathematical physics and engineering. The mathematical modeling of geological folding of rock layers [1], theory of plates and shell [2], waves on a suspension bridge [3], reaction diffusion equation [4], astrophysical narrow convection layers bounded by stable layers [5], viscoelastic and inelastic flows, deformation of beam and plate deflection theory [6–8], and so forth are some of the modeling problems in mathematical physics. The analytical solution of (1) for the arbitrary choice of is difficult and thus we attempt to develop an economical computational method. The existence and uniqueness of the solutions of higher order boundary value problems have been discussed by Agarwal and Krishnamoorthy [9], O’Regan [10], Aftabizadeh [11], and Wei [12]. In the past, the approximate solution for the second, fourth, and/or sixth order two-point boundary value problems has been discussed using homotopy analysis by Liang and Jeffrey [13], reproducing kernel space by Yao and Lin [14], spline solution by Siddiqi and Twizell [15], and the Sinc-Galerkin and Sinc-Collocation methods by Rashidinia and Nabati [16]. The monotone iterative technique and quasilinearization method for the higher order ordinary differential equations have been analysed by Koleva and Vulkov [17]. The geometric mesh technique gains its importance from the theory of electrochemical reaction-convection-diffusion problems in one-dimensional space geometry [18]. The formulation of the geometric mesh finite difference approximations for the two-point singular perturbation