The implicit Euler method or backward Euler method has formulaYi+1 = Yi + hf (ti+1, Yi+1).Apply this method to the model problem y' = λy, (Reλ < 0) to find and plot the region of stability forthis method. Also found the restriction on h for stability if > is real.

Answered: The implicit Euler method or backward…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

The implicit Euler method or backward Euler method has formula
Yi+1 = Yi + hf (ti+1, Yi+1).
Apply this method to the model problem y' = λy, (Reλ < 0) to find and plot the region of stability for
this method. Also found the restriction on h for stability if > is real.

Expert Solution

Introduction

In numerical analysis, the implicit Euler method, also known as the backward Euler method, is a numerical integration scheme for solving ordinary differential equations (ODEs). It is an implicit method that uses backward differences to approximate the derivative in the ODE. In this problem, we will apply the implicit Euler method to the model problem y' = λy, where λ is a complex number with real part less than zero. We will find the region of stability for this method and derive the restriction on h for stability if λ is real.

We will use Python code to plot the region of stability.