Estimation of drifting of snow mass in hilly bound areas by modeling aspect, National conference on snow and avalanche science, Jan , Him Parisar, Sector 37A,Chandigarh, India. Estimation of Shock velocity and pressure for the bullets impacting on Human body, Nov. Estimation of Shock velocity and pressure of detonations and finding their flow parameters, December 17, , University of Pune, India.

Mathematical modeling of an avalanche release and estimation of flow parameters by numerical method. I, Numerical investigation of two-dimensional boundary layer flow over a moving surface, International Journal of Applied Mathematics and Computer Sciences, Vol.

## Ordinary Differential Equations and Integral Equations, Volume 6

Analysis of explosive shock wave and its application in snow avalanche release, International Journal of Computational and Mathematical Sciences, Vol. This is the Euler method or forward Euler method , in contrast with the backward Euler method , to be described below. The method is named after Leonhard Euler who described it in The Euler method is an example of an explicit method.

One often uses fixed-point iteration or some modification of the Newton—Raphson method to achieve this. It costs more time to solve this equation than explicit methods; this cost must be taken into consideration when one selects the method to use.

## Ordinary Differential Equations and Integral Equations, Volume 6

The advantage of implicit methods such as 6 is that they are usually more stable for solving a stiff equation , meaning that a larger step size h can be used. Exponential integrators describe a large class of integrators that have recently seen a lot of development. The Euler method is often not accurate enough. In more precise terms, it only has order one the concept of order is explained below.

This caused mathematicians to look for higher-order methods. This yields a so-called multistep method. Perhaps the simplest is the leapfrog method which is second order and roughly speaking relies on two time values. Almost all practical multistep methods fall within the family of linear multistep methods , which have the form. One of their fourth-order methods is especially popular. A good implementation of one of these methods for solving an ODE entails more than the time-stepping formula. It is often inefficient to use the same step size all the time, so variable step-size methods have been developed.

Usually, the step size is chosen such that the local error per step is below some tolerance level. This means that the methods must also compute an error indicator , an estimate of the local error.

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An extension of this idea is to choose dynamically between different methods of different orders this is called a variable order method. Methods based on Richardson extrapolation , such as the Bulirsch—Stoer algorithm , are often used to construct various methods of different orders. Many methods do not fall within the framework discussed here.

Some classes of alternative methods are:. For applications that require parallel computing on supercomputers , the degree of concurrency offered by a numerical method becomes relevant.

In view of the challenges from exascale computing systems, numerical methods for initial value problems which can provide concurrency in temporal direction are being studied. Numerical analysis is not only the design of numerical methods, but also their analysis. Three central concepts in this analysis are:.

A numerical method is said to be convergent if the numerical solution approaches the exact solution as the step size h goes to 0. The local truncation error of the method is the error committed by one step of the method. That is, it is the difference between the result given by the method, assuming that no error was made in earlier steps, and the exact solution:. Hence a method is consistent if it has an order greater than 0.

The forward Euler method 4 and the backward Euler method 6 introduced above both have order 1, so they are consistent. Most methods being used in practice attain higher order. Consistency is a necessary condition for convergence [ citation needed ] , but not sufficient; for a method to be convergent, it must be both consistent and zero-stable.

A related concept is the global truncation error , the error sustained in all the steps one needs to reach a fixed time t. The global error of a p th order one-step method is O h p ; in particular, such a method is convergent. This statement is not necessarily true for multi-step methods. For some differential equations, application of standard methods—such as the Euler method, explicit Runge—Kutta methods , or multistep methods e.

This "difficult behaviour" in the equation which may not necessarily be complex itself is described as stiffness , and is often caused by the presence of different time scales in the underlying problem. For example, a collision in a mechanical system like in an impact oscillator typically occurs at much smaller time scale than the time for the motion of objects; this discrepancy makes for very "sharp turns" in the curves of the state parameters. Stiff problems are ubiquitous in chemical kinetics , control theory , solid mechanics , weather forecasting , biology , plasma physics , and electronics.

One way to overcome stiffness is to extend the notion of differential equation to that of differential inclusion , which allows for and models non-smoothness. Below is a timeline of some important developments in this field.

## (PDF) Numerical Analysis | John Pryce and G. Berghe - iqegumybiwyf.ml

Boundary value problems BVPs are usually solved numerically by solving an approximately equivalent matrix problem obtained by discretizing the original BVP. This method takes advantage of linear combinations of point values to construct finite difference coefficients that describe derivatives of the function. For example, the second-order central difference approximation to the first derivative is given by:.

One then constructs a linear system that can then be solved by standard matrix methods.

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For instance, suppose the equation to be solved is:. The next step would be to discretize the problem and use linear derivative approximations such as.