Non-uniform Haar Wavelet Method for Solving Singularly Perturbed Differential Difference Equations of Neuronal Variability

A non-uniform Haar wavelet method is proposed on specially designed non-uniform grid for the numerical treatment of singularly perturbed differential-difference equations arising in neuronal variability.We convert the delay and shift terms using Taylor series up to second order and then the problem with delay and shift is converted into a new problem without the delay and shift terms. Then it is solved by using non-uniform Haar wavelet. Two test examples have been demonstrated to show the accuracy of the non-uniform Haar wavelet method. The performance of the present method yield more accurate results on increasing the resolution level and converges fast in comparison to uniform Haar wavelet

Cubic B-spline collocation method for coupled system of ordinary differential equations with various boundary conditions

This paper is concerned with collocation approach using cubic B-spline to solve coupled system of boundary value problems with various boundary conditions. The collocation equations are methodically derived using cubic B splines, for problems with Dirichlet data and an iterative method with assured convergence is described to solve the resulting system of algebraic equations. Problems with Cauchy or mixed boundary condition have been converted into series of Dirichlet problems using the bisection method. Nonlinear problem is linearized using quasilinearization to be handled by our method. Fourth order equation is converted into a coupled second order equations and solved by the proposed method . Several illustrative examples are presented with their error norms and order of convergence.Publisher's Versio

Numerical solution of singularly perturbed problems using Haar wavelet collocation method

Abstract: In this paper, a collocation method based on Haar wavelets is proposed for the numerical solutions of singularly perturbed boundary value problems. The properties of the Haar wavelet expansions together with operational matrix of integration are utilized to convert the problems into systems of algebraic equations with unknown coefficients. To demonstrate the effectiveness and efficiency of the method various benchmark problems are implemented and the comparisons are given with other methods existing in the recent literature. The demonstrated results confirm that the proposed method is considerably efficient, accurate, simple, and computationally attractive

Numerical singular perturbation approaches based on spline approximation methods for solving problems in computational finance

Philosophiae Doctor - PhDOptions are a special type of derivative securities because their values are derived from the value of some underlying security. Most options can be grouped into either of the two categories: European options which can be exercised only on the expiration date, and American options which can be exercised on or before the expiration date. American options are much harder to deal with than European ones. The reason being the optimal exercise policy of these options which led to free boundary problems. Ever since the seminal work of Black and Scholes [J. Pol. Bean. 81(3) (1973), 637-659], the differential equation approach in pricing options has attracted many researchers. Recently, numerical singular perturbation techniques have been used extensively for solving many differential equation models of sciences and engineering. In this thesis, we explore some of those methods which are based on spline approximations to solve the option pricing problems. We show a systematic construction and analysis of these methods to solve some European option problems and then extend the approach to solve problems of pricing American options as well as some exotic options. Proposed methods are analyzed for stability and convergence. Thorough numerical results are presented and compared with those seen in the literature

Numerical singular perturbation approaches based on spline approximation methods for solving problems in computational finance

Philosophiae Doctor - PhDOptions are a special type of derivative securities because their values are derived from the value of some underlying security. Most options can be grouped into either of the two categories: European options which can be exercised only on the expiration date, and American options which can be exercised on or before the expiration date. American options are much harder to deal with than European ones. The reason being the optimal exercise policy of these options which led to free boundary problems. Ever since the seminal work of Black and Scholes [J. Pol. Econ. 81(3) (1973), 637-659], the differential equation approach in pricing options has attracted many researchers. Recently, numerical singular perturbation techniques have been used extensively for solving many differential equation models of sciences and engineering. In this thesis, we explore some of those methods which are based on spline approximations to solve the option pricing problems. We show a systematic construction and analysis of these methods to solve some European option problems and then extend the approach to solve problems of pricing American options as well as some exotic options. Proposed methods are analyzed for stability and convergence. Thorough numerical results are presented and compared with those seen in the literature.South Afric

Implementation of B-splines in a Conventional Finite Element Framework

The use of B-spline interpolation functions in the finite element method (FEM) is not a new subject. B-splines have been utilized in finite elements for many reasons. One reason is the higher continuity of derivatives and smoothness of B-splines. Another reason is the possibility of reducing the required number of degrees of freedom compared to a conventional finite element analysis. Furthermore, if B-splines are utilized to represent the geometry of a finite element model, interfacing a finite element analysis program with existing computer aided design programs (which make extensive use of B-splines) is possible. While B-splines have been used in finite element analysis due to the aforementioned goals, it is difficult to find resources that describe the process of implementing B-splines into an existing finite element framework. Therefore, it is necessary to document this methodology. This implementation should conform to the structure of conventional finite elements and only require exceptions in methodology where absolutely necessary. One goal is to implement B-spline interpolation functions in a finite element framework such that it appears very similar to conventional finite elements and is easily understandable by those with a finite element background. The use of B-spline functions in finite element analysis has been studied for advantages and disadvantages. Two-dimensional B-spline and standard FEM have been compared. This comparison has addressed the accuracy as well as the computational efficiency of B-spline FEM. Results show that for a given number of degrees of freedom, B-spline FEM can produce solutions with lower error than standard FEM. Furthermore, for a given solution time and total analysis time B-spline FEM will typically produce solutions with lower error than standard FEM. However, due to a more coupled system of equations and larger elemental stiffness matrix, B-spline FEM will take longer per degree of freedom for solution and assembly times than standard FEM. Three-dimensional B-spline FEM has also been validated by the comparison of a three-dimensional model with plane-strain boundary conditions to an equivalent two-dimensional model using plane strain conditions

Quartic spline solution of a third order singularly perturbed boundary value problem

Singularly perturbed boundary value problems are solved using various techniques. The spline of degree four is used for the approximate solution of a third order self adjoint singularly perturbed boundary value problem. Convergence analysis is given and the method is proved to be second order convergent. Two examples numerically illustrate the method. References M. Cui and F. Geng, A computational method for solving third order singularly perturbed boundary value problems, Applied Mathematics and Computation, Vol. 198, (2008) pp. 896--903, doi:10.1016/j.amc.2007.09.023 Zengji Du, Singularly perturbed boundary value problem for nonlinear systems, Applied Mathematics and Computation, Vol. 189, (2007) pp. 869--877, doi:10.1016/j.amc.2006.11.167 F. A. Howers, Singular perturbation and differential inequalities, Memoris of the American Mathematical Socity, Providence, Rhode Island, Vol. 168, (1976). Mohan K. Kadalbajoo and Kailash C. Patidar, Numerical soution of singularly perturbed two-point boundary value problem by spline in tension, Applied Mathematics and Computation, Vol. 131, (2002) pp. 299--320, doi:10.1016/S0096-3003(01)00146-1 Petio Kelevedjiev, Existence of positive solutions to a singular second order boundary value problem, Nonlinear Analysis:Theory, Methods and Applications, Vol 50 (8), (2002) pp. 1107--1118, doi:10.1016/S0362-546X(01)00803-3 Arshad Khan, Islam Khan and Tariq Aziz, Sextic spline solution of singularly perturbed boundary value problem, Applied Mathematics and Computation, Vol. 181, (2006) pp. 432--439, doi:10.1016/j.amc.2005.12.059 Manoj Kumar, A fourth order finite differeence method for a class of singular two point boundary value problems, Applied Mathematics and Computation, Vol. 133, (2002) pp. 539--545, doi:10.1016/S0096-3003(01)00255-7 R. K. Mohanty and Navnit Jha, A class of variable mesh spline in compression methods for singularly perturberd two point singular boundary value problem, Applied Mathematics and Computation, Vol. 168, (2005) pp. 704--716, doi:10.1016/j.amc.2004.09.049 J. Rashidinia, R. Mohammadi and M. Ghasemi, Cubic spline solution of singularly perturbed boundary value problem with significant first derivatives, Applied Mathematics and Computation, Vol. 190, (2007) pp. 1762--1766, doi:10.1016/j.amc.2007.02.050, H. G. Roos, M. Stynes and L. Tobiska, Numerical methods for singularly perturbed difference equation, Springer verlag, (1996). Lin Su-rang, Tian Gen-bao and Lin Zong-chi, Singular perturbation of boundary value problem for Quasilinear third order ordinary differential equations involving two small parameters, Applied Mathematics and Mechanics, Vol. 22 (2), (2001) pp. 229--236, doi:10.1023/A:1015553219376 Muhammad I Syam and Basem S. Attili, Numerical solution of singularly perturbed fifth order two point boundary value problem, Applied Mathematics and Computation, Vol. 170 (2005), pp. 1085--1094, doi:10.1016/j.amc.2005.01.003 Ikram A. Tirmizi, Fazal-i-Haq and Siraj-ul-islam, Non-polynomial spline solution of singularly perturbed boundary-value problems, Applied Mathematics and Computation, Vol. 196, (2008) pp. 6--16, DOI: 10.1016/j.amc.2007.05.029,. Wenyan Wang, Minggen Cui and Bo Han, A new method for solving a class of singular two-point boundary value problem, Applied Mathematics and Computation, Vol. 206, (2008) pp. 721--727, doi:10.1016/j.amc.2008.09.01

Total scattering applied to the study of nanomaterials

PhDTotal scattering can be used to study crystalline materials, whose structure presents a periodic arrangement of atoms, as well as disordered materials, such as liquids, glasses or nanomaterials. This thesis work reports three experimental case studies in which different analysis methods were chosen as appropriate on a case-by-case basis. This study demonstrates that total scattering combined with modelling and complementary experimental techniques can guide the understanding of the structure of complex nanostructures. X-ray and neutron total scattering data were collected on multi-walled carbon nanotubes continuously filled with iron and analysed using the program PDFgui for refinement of the pair distribution function and molecular dynamics simulations using the program DL_Poly_4. The analyses show that the iron core is mainly composed of ��-Fe and confirms the dependence of the local ordering on the orientation of the crystallographic axes of iron with respect to the nanowire axis. Prussian blue (Fe4[Fe(CN)6]3 · ��H2O) was synthesised in bulk and nanoparticulate phases using deuterated chemicals; the amount of D2O and H2O in the pores and vacancies, as well as polyvinylpyrrolidone remaining in the nanoparticle samples, were estimated, using an ad hoc modelling procedure of the first few peaks in the neutron PDF function. Models of the structure were refined using the programs PDFgui and RMCProfile. In the last case, a 50Å supercell of the bulk structure with randomly distributed stoichiometric vacancies and D2O and H2O molecules occupying both the pores and the vacancies was used as starting atomic configuration. The CaO/CaCO3 family of materials consists of a series of samples that have undergone carbonation and/or calcination. The X-ray and neutron pair distribution function data were compared to the theoretical PDF of the CaO and CaCO3 phase, generated using the program GULP, that produces PDF functions based on the spectrum of phonon frequencies of the material. The analysis shows that the carbonation is almost completed already after 2 minutes of carbonation and the structure remains stable under further carbonation.EPSRC grant EP/K000128/1