Effects of radiation and heat source on MHD flow of a viscoelastic liquid and heat transfer over a stretching sheet

We study the MHD flow and also heat transfer in a viscoelastic liquid over a stretching sheet in the presence of radiation. The stretching of the sheet is assumed to be proportional to the distance from the slit. Two different temperature conditions are studied, namely (i) the sheet with prescribed surface temperature (PST) and (ii) the sheet with prescribed wall heat flux (PHF). The basic boundary layer equations for momentum and heat transfer, which are non-linear partial differential equations, are converted into non-linear ordinary differential equations by means of similarity transformation. The resulting non-linear momentum differential equation is solved exactly. The energy equation in the presence of viscous dissipation (or frictional heating), internal heat generation or absorption, and radiation is a differential equation with variable coefficients, which is transformed to a confluent hypergeometric differential equation using a new variable and using the Rosseland approximation for the radiation. The governing differential equations are solved analytically and the effects of various parameters on velocity profiles, skin friction coefficient, temperature profile and wall heat transfer are presented graphically. The results have possible technological applications in liquid-based systems involving stretchable materials. © 2005 Elsevier Ltd. All rights reserved

Burgers' Equation and Some Applications

In this thesis, I present Burgers' equation and some of its applications. I consider the inviscid and the viscid Burgers' equations and present different analytical methods for their study: the Method of Characteristics for the inviscid case, and the Cole-Hopf Transformation for theviscid one. Two applications of Burgers' equations are given: one in simple models of Traffic Flow (which have been introduced independently by Lighthill-Whitham and Richards) and another in Coagulation theory (in which we use Laplace Transform to obtain Burgers' equations from the original coagulation integro-differential equation). In both applications we consider only analytical methods

Stochastic Differential Equations with Transformed Anticipating Conditions

Stochastic differential equations can be specified with anticipatory initial value constraints (IVC) which is useful in many applications where future filtration is known. However, such specification leads to different results than those used in the usual Itoˆ’s calculus, and choice of transformations, at IVC, affects the orientation of equivalent isometry moments and related quantities. To solve this problem, the paper analyzes linear stochastic differential equations with anticipatory initial value constraints specified by an exponential transformation. The conditions for general solutions when such function is defined are derived from Itoˆ’s lemma, Taylor, and Fourier Series. Exact solutions are found using the derived conditions as well as those found by other research works. The article also derives numerical scheme for stochastic differential equations with anticipating initial conditions from first principles, Euler-Muruyama and Monte-Carlo methods. The results show that, the Euler-Muruyama method without a Monte-Carlo extension gives reliable numerical solutions for stochastic differential equations with anticipatory initial conditions. The Monte-Carlo extension introduces a slight smoothing effect on the estimated numerical solution

Improving adaptive generalized polynomial chaos method to solve nonlinear random differential equations by the random variable transformation technique

[EN] Generalized polynomial chaos (gPC) is a spectral technique in random space to represent random variables and stochastic processes in terms of orthogonal polynomials of the Askey scheme. One of its most fruitful applications consists of solving random differential equations. With gPC, stochastic solutions are expressed as orthogonal polynomials of the input random parameters. Different types of orthogonal polynomials can be chosen to achieve better convergence. This choice is dictated by the key correspondence between the weight function associated to orthogonal polynomials in the Askey scheme and the probability density functions of standard random variables. Otherwise, adaptive gPC constitutes a complementary spectral method to deal with arbitrary random variables in random differential equations. In its original formulation, adaptive gPC requires that both the unknowns and input random parameters enter polynomially in random differential equations. Regarding the inputs, if they appear as non-polynomial mappings of themselves, polynomial approximations are required and, as a consequence, loss of accuracy will be carried out in computations. In this paper an extended version of adaptive gPC is developed to circumvent these limitations of adaptive gPC by taking advantage of the random variable transformation method. A number of illustrative examples show the superiority of the extended adaptive gPC for solving nonlinear random differential equations. In addition, for the sake of completeness, in all examples randomness is tackled by nonlinear expressions.This work has been partially supported by the Ministerio de Economia y Competitividad grants MTM2013-41765-P.Cortés, J.; Romero, J.; Roselló, M.; Villanueva Micó, RJ. (2017). Improving adaptive generalized polynomial chaos method to solve nonlinear random differential equations by the random variable transformation technique. Communications in Nonlinear Science and Numerical Simulation. 50:1-15. https://doi.org/10.1016/j.cnsns.2017.02.011S1155

Status of the differential transformation method

Further to a recent controversy on whether the differential transformation method (DTM) for solving a differential equation is purely and solely the traditional Taylor series method, it is emphasized that the DTM is currently used, often only, as a technique for (analytically) calculating the power series of the solution (in terms of the initial value parameters). Sometimes, a piecewise analytic continuation process is implemented either in a numerical routine (e.g., within a shooting method) or in a semi-analytical procedure (e.g., to solve a boundary value problem). Emphasized also is the fact that, at the time of its invention, the currently-used basic ingredients of the DTM (that transform a differential equation into a difference equation of same order that is iteratively solvable) were already known for a long time by the "traditional"-Taylor-method users (notably in the elaboration of software packages --numerical routines-- for automatically solving ordinary differential equations). At now, the defenders of the DTM still ignore the, though much better developed, studies of the "traditional"-Taylor-method users who, in turn, seem to ignore similarly the existence of the DTM. The DTM has been given an apparent strong formalization (set on the same footing as the Fourier, Laplace or Mellin transformations). Though often used trivially, it is easily attainable and easily adaptable to different kinds of differentiation procedures. That has made it very attractive. Hence applications to various problems of the Taylor method, and more generally of the power series method (including noninteger powers) has been sketched. It seems that its potential has not been exploited as it could be. After a discussion on the reasons of the "misunderstandings" which have caused the controversy, the preceding topics are concretely illustrated.Comment: To appear in Applied Mathematics and Computation, 29 pages, references and further considerations adde

Iterative Algorithms for Nonlinear Equations and Dynamical Behaviors: Applications

Numerical iteration methods for solving the roots of nonlinear transcendental or algebraic model equations (in 1D, 2D or 3D) are useful in most applied sciences (Biology, physics, mathematics, Chemistry…) and in engineering, for example, problems of beam deflections. This article presents new iterative algorithms for finding roots of nonlinear equations applying some fixed point transformation and interpolation. A method for solving nonlinear systems (in higher dimensions, for multi-variables) is also considered. Our main focus is on methods not involving the equation f(x) in problem and or its derivatives. These new algorithm can be considered as the acceleration convergence of several existing methods. For convergence and efficiency proofs and applications, we solve deflection of a beam differential equation and some test experiments in in Matlab.  Different (real & complex) dynamical (convergence plane) analyzes are also shown graphically. Keywords: nonlinear equations, deflection of beam, iterations, dynamical analysis, applications, 2

New solutions for nonlinear perfect fluids.

Master of Science in Mathematics, Statistics and Computer Science. University of KwaZulu-Natal, Durban 2016.We investigate the Einstein system that governs the evolution of uncharged shear-free spherically symmetric fluids. First we present the Einstein equations for the static spherically symmetric gravitational fields in isotropic coordinates. Also the nonstatic spherically symmetric gravitational fields are studied. We have demonstrated that the fundamental differential equation governing the behaviour of the model is of the Emden-Fowler type. Such equations also arise in applications in Newtonian physics. The field equations governing the gravitational behaviour of the model are generated. We integrate the system of partial differential equations and apply a transformation that reduces the system to a second order ordinary differential equation. To solve the resulting ordinary differential equation we employ the method of characteristics to find different expressions for the gravitational potentials. We employ the method of characteristics to obtain first integrals for the Emden-Folwer type equation. To apply the method, we make use of the associated multipliers which are obtained via the Euler operator acting on the arbitrary multiplier and differential equation. These multipliers can be obtained under the various forms of the arbitrary function representing the gravitational potential under which the equation becomes integrable. Thus expanding the differential equation with the associated multiplier, we can find first integrals by solving the system of partial differential equations. The study is comprised of various forms of the multipliers associated to first integrals of the equation in question

Simplified Expressions of the Multi-Indexed Laguerre and Jacobi Polynomials

The multi-indexed Laguerre and Jacobi polynomials form a complete set of orthogonal polynomials. They satisfy second-order differential equations but not three term recurrence relations, because of the 'holes' in their degrees. The multi-indexed Laguerre and Jacobi polynomials have Wronskian expressions originating from multiple Darboux transformations. For the ease of applications, two different forms of simplified expressions of the multi-indexed Laguerre and Jacobi polynomials are derived based on various identities. The parity transformation property of the multi-indexed Jacobi polynomials is derived based on that of the Jacobi polynomial.ArticleSYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS. 13(0):020 (2017)journal articl

DERIVATIVE UV SPECTROSCOPIC APPROACHES IN MULTICOMPONENT ANALYSIS–A REVIEW

The spectrophotometric multi-component analysis involves spectrum recording and mathematical equations. However, spectral interference poses a major limitation when mixture samples are encountered. To overcome this derivative spectrophotometry (DS) has been introduced for the resolution of overlapping peaks. In this review modified methods like derivative quotient spectra, double divisor ratio spectra derivative method, double divisor means centering of ratio spectra method, derivative subtraction coupled with the constant multiplication method (DS-CM), amplitude subtraction (AS), modified amplitude subtraction (MAS), amplitude factor method (P-Factor), amplitude modulation method (AM), amplitude summation method (A-Sum), simultaneous derivative ratio spectrophotometry (S1DD), derivative compensation ratio via regression equation, differential dual wavelength (D1 DWL), differential derivative ratio (D1DR), successive derivative subtraction method (SDS) and derivative transformation (DT) of derivative spectrophotometry theories and applications are reviewed. These methods were applied to solve different complex pharmaceuticals mixtures. These developed methods were simple and cost-effective

Elzaki transform homotopy perturbation method for partial differential equations

Partial differential equations (PDEs) occur in many applications and play a big role in engineering and applied sciences. Since some PDEs are quite difficult to solve, many new methods are introduced to the academic community. Some of them are homotopy perturbation method, variational iteration method, adomian decomposition method, differential transformation method, ELzaki transform, ELzaki transform homotopy perturbation method (ETHPM) and etc. In this study two methods are considered which is homotopy perturbation method and ELzaki transform. The two methods were introduced and examples were presented to illustrate the efficiency of both methods. It is shown that both methods can be used to solve different types of partial differential equations. Although they can be used to solve PDEs, they have their own limitations. There are certain nonlinear forms of PDEs that are quite difficult to solve using ELzaki transform, and for homotopy perturbation method, the expansion itself sometimes can be quite difficult to solve. Then, the combination of both methods was introduced and the efficiency of the method was shown by solving some applications of partial differential equations. ETHPM was used to solve some gas dynamics and Klein-Gordon equations. The results are compared with previous study to determine the efficiency of the method. The graph of each solution is illustrated by using Mathematica software. From the result, it is shown that ETHPM method produces anticipated exact solutions and the calculations is not that complicated