Geometrical interpretations of Bäcklund transformations and certain types of partial differential equations : a thesis presented in partial fulfilment of the requirements for the degree of Master of Science in Mathematics at Massey University

Page 37 is missing from the original copy.Gauss' Theorema Egregium contains a partial differential equation relating the Gaussian curvature K to components of the metric tensor and its derivatives. Well known partial differential equations such as the Schrödinger equation and the sine-Gordon equation correspond to this PDE for special choices of K and special coördinate systems. The sine-Gordon equation, for example, can be derived via Gauss' equation for K = –1 using the Tchebychef net as a coördinate system. In this thesis we consider a special class of Bäcklund Transformations which correspond to coördinate transformations on surfaces having a specified Gaussian curvature. These transformations lead to Gauss' PDE in different forms and provide a method for solving certain classes of non-linear second order partial differential equations. In addition, we develop a more systematic way to obtain a coordinate system for a more general class of PDE, such that this PDE corresponds to the Gauss equation

ARA-Homotopy Perturbation Technique with Applications

In this study, we propose a novel combination method between the ARA integral transform and the homotopy perturbation approach to solve systems of nonlinear partial differential equations. The difficulty arising in solving nonlinear partial differential equations could simply be overcome by using He’s polynomials during the application of the new method. The proposed technique can provide the solutions of the target problems without pre-assumptions or restrictive constrains in addition to avoiding the round-off errors. The efficiency of the new method is illustrated by applying it to solve different examples of systems of nonlinear partial differential equations. We discuss three interesting applications and solve them by the new approach, called ARA-homotopy perturbation method and get exact solutions, also the results are illustrated in figures

Block methods for direct solution of higher order ordinary differential equations using interpolation and collocation approach

Countless problems in real life situations involve rates of change of one or more independent variables. These rates of change can be expressed in terms of derivatives which lead to differential equations. Conventionally, initial value problems of higher order ordinary differential equations are solved by first reducing the equations to their equivalent systems of first order ordinary differential equations. Then, suitable existing numerical methods for first order ordinary differential equations will be employed to solve the resulting equations. However, this approach will enlarge the equations and thus increases computational burden which may jeopardise the accuracy of the solution. In overcoming the setbacks, direct methods were proposed. Disappointedly, most of the existing direct methods approximate the numerical solution at one point at a time. Block methods were then introduced with the aim of approximating numerical solutions at many points concurrently. Several new block methods using interpolation and collocation approach for solving initial value problems of higher order ordinary differential equations directly were developed in this study to increase the accuracy of the solution. In developing these methods, a power series was used as an approximate solution to the problems of ordinary differential equations of order d. The power series was interpolated at d points before the last two points while its highest derivative was collocated at all grid points in deriving the new block methods. In addition, the properties of the new methods such as order, error constant, zerostability, consistency, convergence and region of absolute stability were also investigated. The developed methods were then applied to solve several initial value problems of higher order ordinary differential equations. The numerical results indicated that the new methods produced better accuracy than the existing methods when solving the same problems. Therefore, this study has successfully produced new methods for solving initial value problems of higher order ordinary differential equations

Solutions of differential equations in a Bernstein polynomial basis

AbstractAn algorithm for approximating solutions to differential equations in a modified new Bernstein polynomial basis is introduced. The algorithm expands the desired solution in terms of a set of continuous polynomials over a closed interval and then makes use of the Galerkin method to determine the expansion coefficients to construct a solution. Matrix formulation is used throughout the entire procedure. However, accuracy and efficiency are dependent on the size of the set of Bernstein polynomials and the procedure is much simpler compared to the piecewise B spline method for solving differential equations. A recursive definition of the Bernstein polynomials and their derivatives are also presented. The current procedure is implemented to solve three linear equations and one nonlinear equation, and excellent agreement is found between the exact and approximate solutions. In addition, the algorithm improves the accuracy and efficiency of the traditional methods for solving differential equations that rely on much more complicated numerical techniques. This procedure has great potential to be implemented in more complex systems where there are no exact solutions available except approximations

Laws of large numbers for mesoscopic stochastic models of reacting and diffusing particles

We study the asymptotic behaviour of some mesoscopic stochastic models for systems of reacting and diffusing particles (also known as density-dependent population processes) as the number of particles goes to infinity. Our approach is related to the variational approach to solving the parabolic partial differential equations that arise as limit dynamics. We first present a result for a model that converges to a classical system of reaction-diffusion equations. In addition, we discuss two models with nonlinear diffusion that give rise to quasilinear parabolic equations in the limit

Solving Systems of Differential Equations of Addition and Cryptanalysis of the Helix Cipher

Mixing addition modulo 2^n (+) and exclusive-or has a host of applications in symmetric cryptography as the operations are fast and nonlinear over GF(2). We deal with a frequently encountered equation (x+y)XOR((x XOR a)+(y XOR b))=c. The difficulty of solving an arbitrary system of such equations -- named differential equations of addition (DEA) -- is an important consideration in the evaluation of the security of many ciphers against differential attacks. This paper shows that the satisfiability of an arbitrary set of DEA -- which has so far been assumed \emph{hard} for large n$-- is in the complexity class P. We also design an efficient algorithm to obtain all solutions to an arbitrary system of DEA with running time linear in the number of solutions. Our second contribution is solving DEA in an adaptive query model where an equation is formed by a query (a,b) and oracle output c. The challenge is to optimize the number of queries to solve (x+y)XOR((x XOR a)+(y XOR b))=c. Our algorithm solves this equation with only 3 queries in the worst case. Another algorithm solves the equation (x+y)XOR(x+(y XOR b))=c$ with (n-t-1) queries in the worst case (t is the position of the least significant `1\u27 of x), and thus, outperforms the previous best known algorithm by Muller -- presented at FSE~\u2704 -- which required 3(n-1) queries. Most importantly, we show that the upper bounds, for our algorithms, on the number of queries match worst case lower bounds. This, essentially, closes further research in this direction as our lower bounds are optimal. We used our results to cryptanalyze a recently proposed cipher Helix, which was a candidate for consideration in the 802.11i standard. We are successful in reducing the data complexity of a DC attack on the cipher by a factor of 3 in the worst case (a factor of 46.5 in the best case)

A family of parametric schemes of arbitrary even order for solving nonlinear models

[EN] Many problems related to gas dynamics, heat transfer or chemical reactions are modeled by means of partial differential equations that usually are solved by using approximation techniques. When they are transformed in nonlinear systems of equations via a discretization process, this system is big-sized and high-order iterative methods are specially useful. In this paper, we construct a new family of parametric iterative methods with arbitrary even order, based on the extension of Ostrowski' and Chun's methods for solving nonlinear systems. Some elements of the proposed class are known methods meanwhile others are new schemes with good properties. Some numerical tests confirm the theoretical results and allow us to compare the numerical results obtained by applying new methods and known ones on academical examples. In addition, we apply one of our methods for approximating the solution of a heat conduction problem described by a parabolic partial differential equation.This research was partially supported by Ministerio de Economia y Competitividad MTM2014-52016-C02-2-P and FONDOCYT 2014-1C1-088 Republica Dominicana.Cordero Barbero, A.; Torregrosa Sánchez, JR.; Vassileva, MP. (2017). A family of parametric schemes of arbitrary even order for solving nonlinear models. Journal of Mathematical Chemistry. 55(7):1443-1460. https://doi.org/10.1007/s10910-016-0723-7S14431460557R. Escobedo, L.L. Bonilla, Numerical methods for quantum drift-diffusion equation in semiconductor phisics. Math. Chem. 40(1), 3–13 (2006)S.J. Preece, J. Villingham, A.C. King, Chemical clock reactions: the effect of precursor consumtion. Math. Chem. 26, 47–73 (1999)H. Montazeri, F. Soleymani, S. Shateyi, S.S. Motsa, On a new method for computing the numerical solution of systems of nonlinear equations. J. Appl. Math. 2012 ID. 751975, 15 pages (2012)J.L. Hueso, E. Martínez, C. Teruel, Convergence, effiency and dinamimics of new fourth and sixth order families of iterative methods for nonlinear systems. J. Comput. Appl. Math. 275, 412–420 (2015)J.R. Sharma, H. Arora, Efficient Jarratt-like methods for solving systems of nonlinear equations. Calcolo 51, 193–210 (2014)X. Wang, T. Zhang, W. Qian, M. Teng, Seventh-order derivative-free iterative method for solving nonlinear systems. Numer. Algor. 70, 545–558 (2015)J.R. Sharma, H. Arora, On efficient weighted-Newton methods for solving systems of nonlinear equations. Appl. Math. Comput. 222, 497–506 (2013)A. Cordero, J.G. Maimó, J.R. Torregrosa, M.P. Vassileva, Solving nonlinear problems by Ostrowski-Chun type parametric families. J. Math. Chem. 53, 430–449 (2015)A.M. Ostrowski, Solution of equations and systems of equations (Prentice-Hall, Englewood Cliffs, New York, 1964)C. Chun, Construction of Newton-like iterative methods for solving nonlinear equations. Numer. Math. 104, 297–315 (2006)A. Cordero, J.L. Hueso, E. Martínez, J.R. Torregrosa, A modified Newton-Jarratt’s composition. Numer. Algor. 55, 87–99 (2010)J.M. Ortega, W.C. Rheinboldt, Iterative solution of nonlinear equations in several variables (Academic, New York, 1970)C. Hermite, Sur la formule dinterpolation de Lagrange. Reine Angew. Math. 84, 70–79 (1878)A. Cordero, J.R. Torregrosa, Variants of Newton’s method using fifth-order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007