There exist
two methods
to find the solution of the differential equation.
How many types of differential equations are there?
We can place all differential equation into
two types
: ordinary differential equation and partial differential equations. A partial differential equation is a differential equation that involves partial derivatives.
How many numerical methods are there?
- Taylor Series method.
- Euler method.
- Runge Kutta methods (RK-2 and RK-4)
- Shooting method.
- Finite difference methods.
What is Runge Kutta 4th order method?
The Runge-Kutta method finds approximate value of y for a given x. Only first order ordinary differential equations can be solved by using the Runge Kutta 4th order method. Below is the formula used to compute next value y
n + 1
from previous value y
n
.
What is the formula of Newton Raphson method?
Therefore it has the equation
y = f ′ ( x n ) ( x − x n ) + f ( x n ) y = f'(x_n)(x – x_n) + f(x_n) y=f′(xn)(x−xn)+f(xn)
.
Which is the most popular Runge-Kutta method?
Runge-Kutta methods of any order can be derived, although the derivation of an order higher than four can become extremely complicated. The most popular method used is
the RK4
, as represented in Eq. (4.1-4).
Which is better Taylor or Runge-Kutta method?
Runge-Kutta method is better
since higher order derivatives of y are not required. Taylor series method involves use of higher order derivatives which may be difficult in case of complicated algebraic equations.
How many steps does the 4th order Runge-Kutta method use?
Explanation: The fourth-order Runge-Kutta method totally has
four steps
. Among these four steps, the first two are the predictor steps and the last two are the corrector steps. All these steps use various lower order methods for approximations.
How can we use Newton Raphson method in real life?
Newton-Raphson method is
extensively used for analysis of flow in water distribution networks
. Several efficient computer programs, using Newton-Raphson method, are also available for analysis of flow in large size networks.
Why does Runge-Kutta work?
The Runge-Kutta Method is a numerical integration technique which
provides a better approximation to the equation of motion
. Unlike the Euler’s Method, which calculates one slope at an interval, the Runge-Kutta calculates four different slopes and uses them as weighted averages.
Why Runge-Kutta method is more accurate than Euler?
The forward Euler method is actually the simplest RK method (1 stage, first order). Higher order accurate RK methods are
multi-stage because
they involve slope calculations at multiple steps at or between the current and next discrete time values.
Which method is best for solving initial value problems?
Some implicit methods have such good stability properties that they can solve stiff initial value problems with step sizes that are appropriate to the behavior of the solution if they are evaluated in a suitable way. The
backward Euler method
and the trapezoidal rule are examples.
What are the disadvantages of Runge-Kutta method?
The primary disadvantages of Runge-Kutta methods are that
they require significantly more computer time than multi-step methods of comparable accuracy
, and they do not easily yield good global estimates of the truncation error
What is the major drawback of Taylor’s series method?
Disadvantages:
Successive terms get very complex and hard to derive
.
Truncation error tends to grow rapidly away from expansion point
.
Almost always not as efficient as curve fitting
or direct approximation.
What is Taylor’s method?
Differential equations – Taylor’s method. Taylor’s Series method. Consider the one dimensional initial value problem y’
= f(x, y)
, y(x
0
) = y
0
where. f is a function of two variables x and y and (x
0
, y
0
) is a known point on the solution curve.
