Example of the Runge – Kutta Method
Example:
Use the 4th order RungeKutta method with h = 0.1 to find the
approximate solution for y(1.1), working to 4 decimal places, for the
initial value problem:
dy/dx = 2xy, y(1) = 1
sol:
We have dy/dx = f(x,y) = 2xy.
If you require the 4th order approximation the formula will be:
y(x0 + h) = y(x0) + (1/6)[w1 + 2w2 + 2w3 + w4]
where:
w1 = h*f(x0.y0) = 0.1(2)(1)(1) = 0.2
w2 = h*f(x0 + h/2, y0 + w1/2) = 0.1(2)(1.05)(1.1) = 0.231
w3 = h*f(x0 + h/2, y0 + w2/2) = 0.1(2)(1.05)(1.1155) = 0.234255
w4 = h*f(x0 + h, y0 + w3) = 0.1(2)(1.1)(1.234255) = 0.2715361
and so:
y(1.1) = 1 + (1/6)[0.2 + 2(0.231) + 2(0.234255) + 0.2715361]
= 1.23367435
We can compare this with the exact solution to the problem.
dy/y = 2x dx
and integrating:
ln(y) = x^2 + C
and y = 1 when x = 1
0 = 1 + C
and so C = 1. Therefore:
ln(y) = x^2 – 1
y = e^(x^21)
When x = 1.1 this gives:
y = e^(1.1^2 1)
= e^0.21
= 1.23367806
which is same as RungeKutta up to the 5th decimal place.
Edited by : Dr. Ashwin I Mehta
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I have asked some of my students to look into the particular topic and shared the details of VISSICOMP TECHNOLOGY. I will be sending you the list of interested students as soon as they confirm.
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Manipal University Jaipur
ONE MORE FEATHER IN OUR CAP
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Thanks for your quick and positive response. I am happy to see your reply. It would be really great thing to connect the IndustryAcademia relationship and feel the gap between Industry and Academic. It will be really a very good platform for young minds. In near future, I will send some more students at your organization definitely and would be happy to invite you in our department for some guest lectures and collaborations to grow together.
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School of Automobile, Mechanical & Mechatronics Engineering (SAMM)
Manipal University Jaipur
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Triangle
 What is the area of triangle DEF?
(A) 23
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(C) 48
(D) 56
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 If AB and BC have equal lengths, what is the value of x?
Quantity A Quantity B
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Programs in Python for BFS
Breadthfirst search (BFS) is an algorithm used for traversing graph data structures. BFS implements a specific strategy for visiting all the nodes (vertices) of a graph. BFS starts with a node, then it checks the neighbours of the initial node, then the neighbours of the neighbours, and so on.
I’ve created a connected graph with 7 nodes and 8 edges. The edges are undirected and unweighted. Distance between two nodes will be measured based on the number of edges separating two vertices. I use the adjacency list to represent the above graph. An effective/elegant method for implementing adjacency lists in Python is using dictionaries. The keys of the dictionary represent nodes, the values have a list of neighbours.
The following screen shot represents representation of the above graph and output shows the result of running Breadth First Search (BFS) algorithm. Can you explain the output? Can you write program in Python ?
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