can some one plz help
how to make trial function for solving equations in collocation and galerkin methods which satisfy the given boundary conditions.
example y"+y=3*square(x) with y(0)=0 and y(2)=3.5.
is there some method of making trial func or we have to make it through intuition..
secondly how to solve equations with neumann conditions y"-(x+1)y=-e^-x(x^2-x+2) subject to neumann cond y'(2)=0,y'(4)=-0.036631
also if someone can giv me the link to and resource where i can get solved examples for my better understanding of these topics.
best regards n hoping to listen soon!
To solve equations using collocation and Galerkin methods, you need to first choose an appropriate trial function that satisfies the given boundary conditions. The trial function represents an approximation of the actual solution. There are no specific rules for constructing trial functions, but they can be devised through intuition, experience, or a systematic approach based on the problem's characteristics.
For the example equation y"+y=3x^2 with y(0)=0 and y(2)=3.5, we can use the collocation or Galerkin method. Here's a step-by-step approach to construct the trial function:
1. Identify the interval: Determine the range over which the equation is to be solved, which in this case is x ∈ [0, 2].
2. Choose a basis function: Select an appropriate set of functions as the basis for the trial function. In collocation or Galerkin methods, commonly used basis functions are polynomials such as power functions.
3. Define the trial function: Multiply each basis function by an unknown coefficient and sum them up. For instance, let's choose trial functions that are linear combinations of two power functions: y = c1 * x + c2 * x^2.
4. Substitute the trial function into the differential equation: Take derivatives of the trial function, substitute them into the equation, and simplify. In this case, y" + y = 3x^2.
5. Apply the boundary conditions: Substitute the boundary values (y(0)=0 and y(2)=3.5) into the trial function. Solve the resulting equations to determine the values of the unknown coefficients (c1 and c2).
Thus, the trial function for the given example is y = c1 * x + c2 * x^2, where c1 and c2 are the coefficients obtained by solving the resulting equations after applying the boundary conditions.
Regarding your second question, solving equations with Neumann conditions requires a similar approach. However, Neumann conditions involve derivatives of the unknown function instead of directly specifying its values. You'll need to incorporate the Neumann conditions into the differential equation before constructing the trial function.
As for additional resources, there are several textbooks and online materials available that provide detailed explanations and solved examples for better understanding of collocation and Galerkin methods. Some recommended resources include:
- "Finite Element Method" by O.C. Zienkiewicz and R.L. Taylor.
- "The Finite Element Method: Linear Static and Dynamic Finite Element Analysis" by Thomas J.R. Hughes.
- "Introductory Finite Difference Methods for PDEs" by D.M. Sloan and C.J. Roberts.
- "The Finite Element Method in Engineering Science" by G.R. Liu and S. S. Quek.
You can also explore online platforms like Khan Academy, Coursera, and YouTube for video tutorials, lectures, and mathematical software demonstrations related to these numerical methods.
Hope this helps, and best of luck with your studies!