Numerical solution of ordinary differential equation
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mathematical solution for the ordinary differential equation and application in aerospace for rocket trajectory by using Runge kutta method .
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Numerical solution of ordinary differential equations GTU CVNM PPT
A ppt on Numerical solution of ordinary differential equations. this PPT contains all gtu content and ideal for gtu students.
Numerical Solution of Ordinary Differential Equations
1. The document discusses numerical methods for solving ordinary differential equations, including power series approximations, Taylor series, Euler's method, and the Runge-Kutta method.
2. It provides examples of using each of these methods to solve sample differential equations and compares the numerical solutions to exact solutions.
3. Truncation errors are defined as errors that result from using an approximation instead of an exact mathematical procedure.
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The document discusses numerical methods for solving differential equations called Runge-Kutta methods. It provides examples of applying the second-order and fourth-order Runge-Kutta methods to solve differential equations. The second-order method uses slopes at the start and middle of each interval to estimate the next value, while the fourth-order method uses slopes at the start, middle, and end of each interval to provide a more accurate estimate. The document also illustrates Heun's method and the second and fourth-order Runge-Kutta methods through examples.
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This document summarizes the Runge-Kutta methods for solving differential equations numerically. It introduces the first, second, third, and fourth order Runge-Kutta methods and provides the equations for calculating each. An example of using the fourth order Runge-Kutta method to solve the differential equation dy/dx=x+y is shown step-by-step. The example calculates the solution to y(0.2) given y(0)=1 using increments of h=0.1.
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Numerical solution of ordinary differential equations GTU CVNM PPT
A ppt on Numerical solution of ordinary differential equations. this PPT contains all gtu content and ideal for gtu students.
Numerical Solution of Ordinary Differential Equations
1. The document discusses numerical methods for solving ordinary differential equations, including power series approximations, Taylor series, Euler's method, and the Runge-Kutta method.
2. It provides examples of using each of these methods to solve sample differential equations and compares the numerical solutions to exact solutions.
3. Truncation errors are defined as errors that result from using an approximation instead of an exact mathematical procedure.
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The document discusses numerical methods for solving differential equations called Runge-Kutta methods. It provides examples of applying the second-order and fourth-order Runge-Kutta methods to solve differential equations. The second-order method uses slopes at the start and middle of each interval to estimate the next value, while the fourth-order method uses slopes at the start, middle, and end of each interval to provide a more accurate estimate. The document also illustrates Heun's method and the second and fourth-order Runge-Kutta methods through examples.
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This document summarizes the Runge-Kutta methods for solving differential equations numerically. It introduces the first, second, third, and fourth order Runge-Kutta methods and provides the equations for calculating each. An example of using the fourth order Runge-Kutta method to solve the differential equation dy/dx=x+y is shown step-by-step. The example calculates the solution to y(0.2) given y(0)=1 using increments of h=0.1.
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The document discusses initial value problems for first order differential equations and Euler's method for solving such problems numerically. It provides an example problem where Euler's method is used to find successive y-values for the differential equation dy/dx = x with the initial condition y(0) = 1. The y-values found using Euler's method are then compared to the actual solution, showing small errors that decrease as the step size h is reduced.
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This document discusses several numerical methods for solving ordinary differential equations (ODEs), including:
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3. Modified Euler's method and the fourth-order Runge-Kutta method, which are single-step methods that do not require computing higher derivatives.
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3) Taylor series allow functions to be approximated by polynomials and are useful in calculus for differentiation, integration, and approximating solutions to problems in physics.
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Numerical solution of ordinary differential equation
- 1. NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATION BY Dixi patel
- 2. INTRODUCTION • A number of numerical methods are available for the solution of first order differential equation of form: • dy/ dx = f(x ,y) • These methods yield solution either as power series or in x form which the values of y can be found by direct substitution, or a set of values of x and y.
- 3. NUMERICAL METHODS • The numerical methods for ODE include: Picard Taylor series Euler series Runge-kutta methods Milne Adams-Bashforth
- 4. RUNGE-KUTTA METHOD • For the first time, the methods were presented by Runge and Kutta two German mathematicians. • These methods were very accurate and efficient and instead direct calculation of higher derivatives only function used for different values.
- 5. WORKING RULE • Working rule to solve the differential equation by runge’s method is given by: k1 = hf (x0,y0) k2 = hf (x0 + 0.5 h, y0 +0.5 k1) k′ = hf (x0 +h, y0 + k1) k3 = hf (x0 +h, y0 + k′) k = 1/6 (k1 +4 k2 + k3 )
- 6. BASIC GRAPH
- 7. RUNGE KUTTA The range kutta is classified as : first order runge kutta dy/dx = f (x ,y) second order runge kutta third order runge kutta fourth order runge kutta
- 8. APPLICATIONS
- 9. APPLICATIONS • If a model rocket is prepared this method can be used for successive iterations of the differential equations for lift off the rocket. • Runge kutta method is used to know the ballistic missile trajectory of a rocket. • In the study of self propelled missiles.
- 10. CONCLUSION • Rocket trajectories are optimized to achieve target by either minimum time or control surfaces or fuel. • First we calculate using the steepest descent method and then it is compared to runge kutta method and results say that the runge kutta method is more efficient and accurate . • But takes time compared to others .
- 11. THANK YOU