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This paper deals with the design and analysis of a robust numerical scheme based on an improvised quartic B-spline collocation (IQBSC) method for a class of nonlinear derivative dependent singular boundary value problems (DDSBVP). The convergence analysis of the method is studied by means of Green's function approach. It should be pointed out that the numerical order of convergence of standard quartic B-spline collocation (SQBC) scheme for second-order boundary value problems (BVPs) is four, however, our proposed IQBSC method is shown to be sixth order convergence. To illustrate the applicability and accuracy of the method, we consider eight test problems. The obtained results are compared to those from some existing numerical schemes in order to show the advantage of present method. It is shown that the rate of convergence of present numerical scheme is higher than that of some of existing numerical methods. The CPU time of the present numerical method is provided. [ABSTRACT FROM AUTHOR]

A new symmetric six-step method with improved phase properties is introduced, for the first time in the literature, in this paper. The proposed method: (1) is a symmetric nonlinear six-step method, (2) is of three-stages, (3) is of twelfth algebraic order, (4) has vanished the phase-lag and (5) has vanished the derivatives of the phase-lag up to order five. For the new symmetric six-step method we present a detailed theoretical, numerical and computational analysis which consists of: (a) the development of the new three-stages symmetric six-step method, (b) the computation of the local truncation error of the new scheme, (c) the comparative error analysis of the new algorithm with other schemes of the same family: (i) the classical algorithm of the family (i.e. the method with constant coefficients), (ii) the recently proposed method of the same family with vanished phase-lag and its first derivative, (iii) the recently proposed method of the same family with vanished phase-lag and its first and second derivatives, (iv) the recently proposed method of the same family with vanished phase-lag and its first, second and third derivatives and finally (v) the recently proposed method of the same family with vanished phase-lag and its first, second, third and fourth derivatives, (d) the stability and the interval of periodicity analysis for the new developed method and finally (e) the study of the accuracy and computational effectiveness of the newproposed method for the solution of the Schrödinger equation. The theoretical, numerical and computational analysis of the newobtained three-stages symmetric six-step method show the efficiency of the new algorithm compared with other known or recently developed schemes of the literature. [ABSTRACT FROM AUTHOR]

In this paper we will develop an explicit fourth algebraic order four-step method with phase-lag and its first and second derivatives vanished. The comparative error and the stability analysis of the above mentioned paper is also presented. The new obtained method is applied on the resonance problem of the Schrödinger equationIn order in order to examine its efficiency. The theoretical and the computational results shown that the new obtained method is more efficient than other well known methods for the numerical solution of the Schrödinger equation and related initial-value or boundary-value problems with periodic and/or oscillating solutions. [ABSTRACT FROM AUTHOR]

In this paper we introduce a new explicit hybrid Numerov-type method. This method is of fourth algebraic order and has phase-lag and its first two derivatives equal to zero. We present a stability analysis and an error analysis based on the radial Schrödinger equation. Finally we apply the new proposed method to the resonance problem of the radial Schrödinger equation and we present the final conclusion based on the theoretical analysis and numerical results. [ABSTRACT FROM AUTHOR]

A mathematical model of mediated bioelectrocatalysis is restudied in this paper. Here He's Homotopy perturbation method is implemented to give approximate and analytical solutions of steady-state non-linear reaction diffusion equation containing a non-linear term related to Michaelis-Menten kinetics of the enzymatic reaction. Approximate analytical expressions for mediator concentration and current have been derived for all values of saturation parameter α and reaction diffusion parameter k for the various types of boundary conditions. The Homotopy perturbation method which produces the solutions in terms of convergent series, requiring no linearization or small perturbation. These analytical results are compared with numerical results (Matlab program) and are found to be in good agreement. [ABSTRACT FROM AUTHOR]

In this work, for the first time in an article, we present in a comprehensive way the explicit advanced step-point (EAS) methods. The EAS methods is a family of methods designed for the numerical solution of non-stiff and mildly stiff initial value problems (IVPs) and comprises three distinct schemes: EAS1, EAS2 and EAS3. A thorough theoretical analysis of the EAS family of predictor–corrector methods is presented in terms of their accuracy and stability characteristics and requirements, as well as the rationale for creating the three distinct schemes mentioned above. In this paper we also examine in detail one of the three schemes, the EAS1 methods. EAS1 are assessed for the very first time, are meticulously studied and their superior regions of absolute stability are presented. Furthermore the computational efficiency of EAS1 is examined and comparative numerical results are presented with the use of a variable step, variable order EAS1 code. The numerical results provide good evidence that EAS1 could be seen as superior to the well established Adams methods for the numerical solution of mildly stiff initial value problems. [ABSTRACT FROM AUTHOR]