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Computer solution of non-linear integration formula for solving initial value problems

Abdul Razak bin Yaakub

# Computer solution of non-linear integration formula for solving initial value problems

## by Abdul Razak bin Yaakub

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Published .
Written in English

Edition Notes

Thesis (Ph.D.) - Loughborough University, 1996.

 ID Numbers Statement by Abdul Razak bin Yaakub. Open Library OL19039080M

Now we substitute the value $$C=2$$ into Equation. The solution to the initial-value problem is $$y=3e^x+\frac{1}{3}x^3−4x+2.$$ Analysis. The difference between a general solution and a particular solution is that a general solution involves a family of functions, either explicitly or implicitly defined, of the independent variable. Use DSolve to solve the differential equation for with independent variable: The solution given by DSolve is a list of lists of rules. The outermost list encompasses all the solutions available, and each smaller list is a particular solution.

2 Boundary Value Problems If the function f is smooth on [a;b], the initial value problem y0 = f(x;y), y(a) given, has a solution, and only one. Two-point boundary value problems are exempli ed by the equation y00 +y =0 (1) with boundary conditions y(a)=A,y(b)=B. An important way to analyze such problems is to consider a family of solutions of File Size: KB.   OSLO is and Silverlight class library for the numerical solution of ordinary differential equations (ODEs). The library enables numerical integration to be performed in C#, F# and Silverlight applications. OSLO implements Runge-Kutta and back differentiation formulae (BDF) for non-stiff and stiff initial value problems. We wrote this library, in collaboration with Moscow State [ ].

In this section we solve separable first order differential equations, i.e. differential equations in the form N(y) y' = M(x). We will give a derivation of the solution process to this type of differential equation. We’ll also start looking at finding the interval of validity for the solution to a differential equation. Types of Solutions. "Elementary Differential Equations and Boundary Value Problems by W. E. Boyce and R. C. DiPrima from John Wiley & Sons" is a good source for further study on the subject. For example, consider this first order non-linear differential equation say, y prime squared + 1 = 0, right? As you know well, any real.

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### Computer solution of non-linear integration formula for solving initial value problems by Abdul Razak bin Yaakub Download PDF EPUB FB2

This thesis is concerned with the numerical solutions of initial value problems with ordinary differential equations and covers single step integration methods. focus is to study the numerical the various aspects of Specifically, its main methods of non-linear integration formula with a variety of means based on the Contraharmonic mean (C˳M) (Evans and Yaakub []), the Centroidal mean (C Author: Abdul Razak Bin Yaakub.

This thesis is concerned with the numerical solutions of initial value problems with ordinary differential equations and covers single step integration methods. focus is to study the numerical the various aspects of Specifically, its main methods of non-linear integration formula with a variety of means based on the Contraharmonic mean (C.M) (Evans and Yaakub []), the Centroidal mean (C.M Author: Abdul Razak Bin Yaakub.

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The combination (–2) is referred to as an initial value problem, and our goal is to devise both analytical and numerical solution strategies. A diﬀerential equation is called autonomous if the right hand side does not explicitly depend upon the time variable: du dt = F(u).

() All autonomous scalar equations can be solved by direct integration. This chapter discusses the numerical treatment of singular/discontinuous initial value problems.

The mathematical formulation of physical phenomena in simulation, electrical engineering, control theory, and economics often leads to an initial value problem in which there is a pole in the solution or a discontinuous low order derivative.

using an iteration method, solutions are easily ob-tained. Moreover, some examples are shown exact solutions. Keywords: initial value problem, successive approxima-tion, di erential equation, volterra integral equation, laplace transform 1 Introduction Finding exact solutions of nonlinear initial value.

Systems of Non-Linear Equations Newton’s Method for Systems of Equations It is much harder if not impossible to do globally convergent methods like bisection in higher dimensions. A good initial guess is therefore a must when solving systems, and Newton’s method can be used to re ne the guess.

The rst-order Taylor series is f xk + x ˇf xk File Size: KB. For simple differential equations, it is possible to ﬁnd clo sed form solutions. For example, given a function g, the general solution of the simplest equation Y′(t) = g(t) is Y(t) = Z g(s)ds+c with can arbitraryintegrationconstant.

Here, R g(s)dsdenotes anyﬁxed antideriva-tive of Size: 1MB. solving differential equations. With today's computer, an accurate solution can be obtained rapidly.

In this section we focus on Euler's method, a basic numerical method for solving initial value problems. Consider the differential equation: The first step is to convert the above second-order ode into two first-order ode.

This is a standard. Boundary value problems (BVPs) are usually solved numerically by solving an approximately equivalent matrix problem obtained by discretizing the original BVP. The most commonly used method for numerically solving BVPs in one dimension is called the Finite Difference Method.

. In numerical analysis, the shooting method is a method for solving a boundary value problem by reducing it to the system of an initial value y speaking, we 'shoot' out trajectories in different directions until we find a trajectory that has the desired boundary value.

The following exposition may be clarified by this illustration of the shooting method. 54 Boundary-ValueProblems for Ordinary Differential Equations: Discrete Variable Methods with g(y(a), y(b» = 0 (b) Ifthe number of differential equations in systems (a) or (a) is n, then the number of independent conditions in (b) and (b) is n.

In practice, few problems File Size: 1MB. Application: Newton’s Law of Motion Newton’s Law of Motion is F = ma Acceleration is the time derivative of velocity, so dv dt = a and dv dt = F m If F(t) and v(0) are known, we can (at least in principle) integrate the preceding equation to ﬁnd v(t) NMM: Integration of ODEs page 3File Size: KB.

In this case, the only free variables in [(x,0), (x, ) - 3] are C1 and C2, so one can see why solve solves for them, but I've seen that solve Does The Right Thing even when there are other free variables in the equations of initial/boundary conditions Neat.

On the other hand, I've spent some time trying to tell solve that I wanted said equations solved with respect. Pressing the Solve button sets Solver going on its search for a solution.

After a few seconds, Solver presents the results shown in Figure Figure Final solution to initial value problem. As you can see, all the residuals are 0 and the upper table now contains the solution to this initial value.

For a linear integral equation, you can use the same collocation method, but the integralEquations will be linear, so you can be sure of finding a solution simply by using Solve, or reformulate as a matrix problem and use LinearSolve.

A clever method for solving differential equations (DEs) is in the form of a linear first-order equation. This method involves multiplying the entire equation by an integrating factor.

A linear first-order equation takes the following form: To use this method, follow these steps: Calculate the integrating factor. The Numerical Methods for Linear Equations and Matrices • • deterministic (i.e. given the same initial state, the computer will always arrive at the same answer), a large We now turn to the solution of linear algebraic equations and problems involving matricesFile Size: KB.

Numerical Methods for Ordinary Differential Equations is a self-contained introduction to a fundamental field of numerical analysis and scientific computation. Written for undergraduate students with a mathematical background, this book focuses on the analysis of numerical methods without losing sight of the practical nature of the subject.

Differential Equation Calculator The calculator will find the solution of the given ODE: first-order, second-order, nth-order, separable, linear, exact, Bernoulli, homogeneous, or inhomogeneous. Initial conditions are also supported. A particular solution is a solution corresponding to a specific value of the integration constants.

For example, the function y = x2/2 is a particular solution to the equation, dy/dx – x = 0. A general solution for this equation would be y = x2/2 + C, where C is an arbitrary integration Size: KB.The General Initial Value Problem. We are trying to solve problems that are presented in the following way: dy/dx=f(x,y); and y(a) (the inital value) is known, where f(x,y) is some function of the variables x, and y that are involved in the problem.

Examples of Initial Value Problems.This book is aimed at students who encounter mathematical models in other disciplines. It assumes some knowledge of calculus, and explains the tools and concepts for analysing models involving sets of either algebraic or 1st order differential equations/5(42).