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Solution of linear initial value problems on a hypercube

Beny Neta

Solution of linear initial value problems on a hypercube

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Published by Naval Postgraduate School, Available from National Technical Information Service in Monterey, Calif, Springfield, Va .
Written in English

    Subjects:
  • BOUNDARY VALUE PROBLEMS

  • About the Edition

    There are many articles discussing the solution of boundary value problems on various parallel machines. The solution of initial value problems does not lend itself to parallelism, since in this case one uses methods that are sequential in nature. The authors develop a parallel scheme for initial value problems based on the box scheme and a modified recursive doubling technique. Fully implicit Runge Kutta Methods were discussed by Jackson and Norsett (1986) and Lie (1987). Lie assumes that each processor of the parallel computer having vector capabilities. (kr)

    Edition Notes

    Other titlesNPS-53-89-001.
    StatementBeny Neta
    ContributionsNaval Postgraduate School (U.S.). Dept. of Mathematics
    The Physical Object
    Pagination9 p. ;
    ID Numbers
    Open LibraryOL25496906M

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Solution of linear initial value problems on a hypercube by Beny Neta Download PDF EPUB FB2

To solve initial value problems for linear differential equations with constant coefficients. The Laplace transform is useful in solving these differential equations because the transform of f ' is related in a simple way to the transform of f, as stated in Theorem File Size: 1MB.

Schemes for the solution of linear initial or boundary value problems on a hypercube were developed by Katti and Neta [1] and tested and improved by Lustman, Neta and Katti [2]. In this paper there is developed and tested a parallel scheme for the solution of linear systems of ordinary initial value problems based on the box scheme and a modified recursive doubling technique.

The box scheme may be replaced by any stable integrator. The algorithm can be modified to solve boundary value problems. Software for both problems is available upon by: 4.

Consider the linear system $$ \frac{dY}{dt} = \begin{pmatrix} 1 & -1 \\ 1 & 3 \\ \end{pmatrix} Y $$ (a) Show that the function $$ Y(t) = \begin{pmatrix} te^{2t} \\ -(t + 1)e^{2t}\\ \end{pmatrix} $$ is a solution to the differential equation.

I verified this without trouble. The second part I. Numerical Solution of Initial Value Problems. In particular, if, the IVP above is called autonomous and if g(y) = ky where k is a constant, the IVP is linear.

We assume that a unique solution exists and denote that solution by y e (t). So, for from now on. In the field of differential equations, an initial value problem (also called a Cauchy problem by some authors [citation needed]) is an ordinary differential equation together with a specified value, called the initial condition, of the unknown function at a given point in the domain of the physics or other sciences, modeling a system frequently amounts to solving an initial value.

initial-value problems is beyond the scope of this course. Exercises 1. (a) Show that each Solution of linear initial value problems on a hypercube book of the one-parameter family of functions y = Ce5x is a solution of the differential equation y0 − 5y =0.

(b) Find a solution of the initial-value problem y0 −5y =0,y(0) = 2. (a) Show that each member of the two-parameter family of functions. solve initial value problem (linear differential equation) 1. Ordinary Differential Equations ODEs Solve the Initial Value Problem.

Initial value problem for a linear system. Create triangle solving problems Ionizing radiation in thermal radiation How many envelops were used in the scene where Harry receives his Hogwarts letter?.

Prentice J () The RKGL method for the numerical solution of initial-value problems, Journal of Computational and Applied Mathematics,(), Online publication date: Mar Hashemian A and Shodja H () A meshless approach for solution of Burgers' equation, Journal of Computational and Applied Mathematics,( Problems for Ordinary Differential Equations INTRODUCTION The goal of this book is to expose the reader to modern computational tools for solving differential equation models that arise in chemical engineering, e.g., diffusion-reaction, mass-heattransfer, and fluid flow.

The emphasis is placedFile Size: 1MB. the initial value problem may fail to have a unique solution over any time interval if this initialvalue is imposed. Example Considerthe initialvalue problem u0.t/D p u.t/ withinitialcondition u.0/D 0: The function f.u/ D p u is not Lipschitz continuous near u D 0 since f0.u/ D p u/!1as u.

Size: KB. InitialValueProblems Aswehaveseen, mostdifferentialequationshavemorethanonesolution. Fora first-orderequation File Size: KB. Initial Value Problems When we solve differential equations, often times we will obtain many if not infinitely many solutions.

For example, consider the differential equation $\frac{dy}{dx} = y$. so we have found the general solution of the differential equation (with a 0 instead of B, and a 1 /2 instead of A). The series solutions method is mainly used to find power series solutions of differential equations whose solutions can not be written in terms of familiar functions such as polynomials, exponential or trigonometric functions.

Keywords: initial value problem, successive approxima-tion, di erential equation, volterra integral equation, laplace transform 1 Introduction Finding exact solutions of nonlinear initial value prob-lems (IVPs) is a goal for mathematicians, engineers, and scientists.

Math A Spring Week 9 Solutions Burden & Faires § 1a, 3a, 5, 7 Burden & Faires § 1ab, 2b, 5a, 7 Burden & Faires § 3a, 5ac, 7 Burden & Faires § The Elementary Theory of Initial-Value Problems 1.

Use Theorem to show that the following initial-value problem has a unique solution, and find the solution. Size: KB. In this work we review the present status of numerical methods for partial differential equations on vector and parallel computers.

A discussion of the relevant aspects of these computers and a brief review of their development is included, with particular attention paid to those characteristics that influence algorithm selection.

Both direct and iterative methods are given for elliptic Cited by: The initial value problem solution of the nonlinear shallow water-wave equations is developed under initial waveforms with and without velocity.

W e present a solution method based on a hodograph. Solution (c): The phase-line shows that if y(0) = −1 then y(t) →0 as t →∞. (2) Solve (possibly implicitly) each of the following initial-value problems. Identify their intervals of definition. (a) dy dt + 2ty 1+t2 = t2, y(0) = 1. Solution: This equation is linear and is already in File Size: KB.

Express the solution of the following initial value problem in terms of a convolution integral: y(4) − y= g(t); y(0)= y Recall that we can write any single linear homogeneous equation of order n into a 1st order system consisting of n equations.

Show that the Wronskian of the File Size: 96KB. The Existence and Uniqueness (of the solution of a second order linear equation initial value problem) A sibling theorem of the first order linear equation Existence and Uniqueness Theorem Theorem: Consider the initial value problem y″ + p(t) y′ + q(t) y = g(t), y(t0) = y0, y′(t0) = y′ Size: KB.

Differential Equations with Boundary-Value Problems (8th Edition) Edit edition. Problem 9E from Chapter In Problems 1–10 solve the given differential equation by us %(41).

Why is resulting solution optimal. Any feasible solution satisfies system of equations in tableaux. • In particular: Z = – S C – 2 S H • Thus, optimal objective value Z* since S C, S H 0. • Current BFS has value optimal.

basis = {A, B, S M} S C = S H = 0 Z = B = 28 A = 12 S M = maximize Z subject to the. Numerical Initial Value Problems in Ordinary Differential Equations August August Read More. Author: C. William Gear. Consider the following system of linear differential equations. x0 = 6 5 2 −3 x (a) Find the special fundamental matrix Φ(t) which satisfies Φ(0) = I.

(b) Solve the following initial value problem using the fundamental matrix found in (a). x0 = 6 5 2 −3 x, x(0) = 1 −2 (c) Draw the phase portrait of the given system. Size: KB. 4 CHAPTER 9. THE INITIAL VALUE PROBLEM A good way to visualize the LTE is to recognize that at each step, (tn,yn) sits on some solution curve yn(t) that satisfies the differential equation y0(t) = f(t,y(t)).

With each step we jump to a new solution curve, and the size of the jump is the LTE. (See Figure ). Solve the initial value problem yy' + x = x2 y2 with y(1) = - 3.

To solve this, we should use the substitution After the substitution from the previous part, we obtain the following linear differential equation in x, u, u' The solution to the origional initial value problem is described by the following equation in x,y.

I have the following initial value problem in my math book: Find the solution for the initial value problem 2y'' - 3y' + y = 0, y(0) = 2, y'(0) = 1/2 Then determine the maximum value of the solution and also find the point where the solution is zero.

I have been able to find that the solution to the problem is y = -(e^t) + 3e^(t/2) and the solution in the back of the book confirms that.

Introduction to Differential Equations Part 2: Initial value problems. Our mathematical model describing the spread of the rumor consists of two parts. The first is the differential equation. dS/dt = k S (M - S). The second part is the condition that two students knew the rumor at the beginning.

S(0) = 2. Solve each of the following initial value problems and plot the solutions for several values of y0. Then describe in a few words how the solutions resemble, and differ from, each other.

dy/dt = −y + 5, y(0) = y0 Ive just started differential equations, and its really hard. Mostly bcuz my teachers method of teaching is kinda scattered. We just have to remind ourselves that the Laplace transform of the unit step function-- I'll put the pi there, just 2 pi times f of t minus 2 pi-- I should put as the step function of t-- is equal to e to the minus 2 pi s times the Laplace transform of just-- or let me just write it this way-- times the Laplace transform of f of t.

Express the solution of the given initial value problem as a sum of two oscillations as in Eq. (8). Throughout, primes denote derivatives with respect to time t. x′′+25x =10cos4t; x(0) =25, x′(0) =10 First the complementary solution will be found, then the particular solution will be found,File Size: KB.

Find an explicit solution to this initial value problem. Solution The function f(x,y) = 3 √ y is continuous in the entire real plane, but the y-derivative D yf(x,y) = 1 3 (y)−2/3 is not continuous at (0,0).

Theorem 1 from section does not guarantee existence of a solution with y(0) = File Size: 54KB. the initial value problem: existence, uniqueness, extensibility, dependence on initial conditions.

Furthermore we consider linear equations, the Floquet theorem, and the autonomous linear ow. Then we establish the Frobenius method for linear equations in the com-plex domain and investigate Sturm{Liouville type boundary value problems. Calculus, PureMath, Initial value problem, linear ordinary differential equation, linear ODEs, fundamental matrix, homogeneous ordinary differential equation, homogeneous solution Downloads LinODE.m ( KB) - Mathematica package.

Abstract. Let A be a symmetric n by n matrix where B = I – A is the corresponding Jacobi matrix: We assume B has been re-ordered using the red/black ordering so that.

Using the Singular Value Decomposition of M, we are able to derive an equation for computing the eigenvalues of the two-parameter SOR iteration matrix Lω 1,ω addition, we give a method for computing the spectral.

By (6), Eq. 7 becomes substituting the initial condition and the values find we obtain the solution y (x) = x 4-x CONCLUSION. In the discussion it was shown that, with the proper use of the taylor series method, it is possible to obtain an analytic solution to a class of singular initial value problems, homogeneous or inhomogeneous.

Total 7 Questions have been asked from Initial And Boundary Value Problems topic of Differential equations subject in previous GATE papers. The solution of the initial value problem d y d x =-2 x y; (Linear And Nonlinear). Chapter 3. Second Order Linear Equations In each of Problems 9 through 16 find the solution of the given initial value problem.

Sketch the graph of the solution and describe its behavior as t increases. −2y = 0, y(0) = 1, y. PROBLEMS In each of Problems 1 through 6 find theWronskian of the given pair of Size: KB. (b) Describe the behaviour of the solution corresponding to the initial value.

*Consider the initial value problem: Find the coordinates of the first local maximum point of the solution for. *Consider the initial value problem: Find the value of for which File Size: KB.

Superb introduction devotes almost half its pages to numerical methods for solving partial differential equations, while the heart of the book focuses on boundary-value and initial-boundary-value problems on spatially bounded and on unbounded domains; integral transforms; uniqueness and continuous dependence on data, first-order equations, and more.5/5(1).In this work, we propose a novel computational algorithm for solving linear and nonlinear initial value problems by using the modified version of differential transform method (DTM), which is called the projected differential transform method (PDTM).

The PDTM can be easily applied to the initial value problems with less computational by: y =(1+x)α satisfies the initial value problem (1+x)y′=αy, y(0) =1. (b) Show that the power series method gives the binomial series in (12) as the solution of the initial value problem in part (a), and that this series converges if x File Size: KB.