Late Chandrashekhar ji Purva Pradhanmantri Smarak Mahavidyalaya Seorai Ghazipur.
The Euler method is a basic numerical algorithm for solving ordinary differential equations (ODEs) that occur in different scientific and engineering disciplines. The paper describes a detailed study and application of the Euler method, with a specific focus on solving first-order ODEs. The Euler method is studied with respect to its algorithmic steps, computational complexity, and drawbacks. The technique approximates the solution of ODEs by breaking down the problem into small steps and iterating through them, providing a straightforward and easy-to-understand method for solving initial value problems. Although it is computationally efficient, the precision of the method relies on the step size, with smaller steps providing better approximation but at greater computational expense. The paper also contrasts the Euler method with other higher-order methods, including Runge-Kutta, and discusses its trade-offs. The application of the Euler method is illustrated by a number of example problems, and the outcome is examined both in terms of accuracy and computational efficiency. Lastly, the paper concludes with some recommendations on how the Euler method should be applied in different situations based on its strengths and limitations
One the simplest and oldest methods for approximating differential Equations as Euler method. The Euler method is a first order method which means that the local error is proportional global error is proportional to the step size. Euler's Method is a just another techniques use to analyse a differential equation which uses the idea linearity or linear approximation where we use small tangent line over the short distance to approximation the solution to an initial value problem.
Discription Of Method
Consider the problem of caculating the step shape Shape of an unknown curve which starts at a given differential equation . Here is a differential equation can be through as a formula by which the slope of the tangent line to the curve Can be computed at any point has been calculated. The idea is that while the curve is initially unknown its starting point , which we denote by Ao , is known. Then from the differential Equations the curve A0 can be Computed, and so the tangent line. Take a small step along that tangent line up to a point A , Along this small step the slope does not change too much . So A, will be close to the curve , if we pretend the A1 is still on the curve the same reasoning as for the point Ao Above can be used . After several steps a polygonal curve (A0, A1 , A2...) is computed . In general this curve does not diverge too for from the original unknown curve and the error between the two curves Can be made small if the step size is small enough and the internal of computation is finite.
First Order Process
When given the value for to and y(t0) , and the derivative of y is a given function of T and y denoted as :
y'(t) = f[t, y(t)]
Begin the process by setting :
Y0 = y(t)
Next choose a value H for of the size and of every step and set :
Dn= t0 + nh (or equivalently)
Tn+1 = Tn+1
Now, One step of the Euler method from tn T0 Tn+1
Yn+1 = Yn + nf(tn+Yn)
The value of Yn is an opproximation of the solution to the ordinary differential equation at the time Tn.
Yn =y(t)
The Euler Method is explicit. ie. the solution Yn+1 is an exploit function of Yi for i-<n.
Higher order Process
While the Euler method integrates a first order ODE of order N can represented as a system of the first order differential equation . When given the Ode of order N defined as :
y(t) = f[t, y(t), y'(t)....y (t)]
We implement the following formula until we reach the approximation of solution to the ODE time
These first order system can be handled by euler method or infact by any other scheme for first order system.
Implementation Of Method
Eulers method is algorithm for approximating the solution to the value problem by following the tangent lines while we take horizontal step across the t-Axis .
If we wish to approximate ylt).
As an example we will use Euler Method to solve the Equation form page
dy (t) = ay (t) dt
With the initial condition y(0) = 5 billion cells and Growth the growth parameter a = 0.2 per hour. As a first step create a file named Euler method with top - level function and helper function. Function res = Euler () T (1) = 0; Y(1)= S r= ratefunc [T(1), Y(1) ] end
Function res = rate - func (t, y)
a = 0.2 dy * dt = a*y res= dy /dt:
End
In Euler we initialize the initial conditions and then call rate-func, so called because it computes the rate of growth in the population. After testing functions, we can add code to Euler to compute these difference to Equations
Where, r is the rate of growth computed by rate function, Listing 9.1 has code we need.
Listing 9.1; A function implementing Euler's method.
Function res = Euler () T (1) = 0: Y (1) = 5; dt= 0.1; far i = 1:40 r= rate - func [T(i), Y(i)]; T(i + 1) = T(i) + dt; Y(i+ 1) = y(i) + r * dt; End Plot (T,Y) end. Before the loop we create two vector, T and Y, and set the first element of each with the Initial conditions; dt, which is the size of the time, steps is 0.1 hour's . Inside the loop , We compute growth rate the based on current time ,T(i), and population. Y(i) , You might notice that the rate depends only on population but we pass time as an input variable any way, for reason you , will see soon. After computing the growth rate, we add an element boths T and Y. Then ,when the loop exist , we plot Y as a function of T . If you run the code you should get a plot of the population over time as shown in figure. 9.1
figure 9.1 solution to a simple differential Equation by Euler's Method , As you can see the population doubles in little less than 4 Hours
Euler's Method Formula
Euler's method paint Can be used to estimate point on the solution to Y’=f(n0,y0) which posses through any given initial point (no, Yo) in the plane. Using a linear approximation to the curve at that point the value of the solution at a nearby point n1=n0+h is approximately , y = y 1= y0 + hf’ (n0,y0)
CONCLUSION
This examination and implementation of the Euler method in solving ordinary differential equations reflects its ease and usefulness in dealing with initial value problems. Given its practicality, the Euler method is rather simple to use which makes it reliable for quick approximations of ODE solutions that cannot easily be solved analytically. Yet there are some significant weaknesses, including, but not limited to, relatively low accuracy of the result, especially for stiff equations which need tight precision. The precision of the result obtained using the Euler method is quite sensitive to the value of the step size. Lesser values of step size provides more accurate results, but increases the amount of resources needed for computation. The method also has a tendency to add on error, most especially in long term integrations where the solution tends to depart from the real state of the system with time. Despite all the drawbacks, the Euler method allows the easier approximation of more advanced methods, for example, Runge-Kutta methods that require almost the same amount of resources but are more accurate. In most cases, the Euler method is a practical concept when teaching students on solving numerical problems involving differential equations, whereas other advanced methods would be used when simulating real life problems requiring precision.
REFERENCES
Supriya Tiwari*, Renu Singh, Priyanka Tiwari, Lovely Nigam, Pooja Shahoo, Analysis And Implementation of Euler Method for Solving Ordinary Differential Equation, Int. J. of Pharm. Sci., 2025, Vol 3, Issue 4, 528-534 https://doi.org/10.5281/zenodo.15148232
10.5281/zenodo.15148232
