Multi-point Boundary Value Problems Using Finite Difference Method

In summary, "Multi-point Boundary Value Problems Using Finite Difference Method" discusses the application of the finite difference method to solve boundary value problems characterized by multiple boundary conditions. The paper outlines the mathematical formulation of these problems, presents the discretization process using finite differences, and illustrates the implementation of numerical algorithms to obtain approximate solutions. It emphasizes the significance of addressing multi-point conditions effectively and compares the performance of various finite difference schemes, highlighting their accuracy and convergence properties. Additionally, the study includes examples to demonstrate the practical application of the method in solving complex differential equations.
  • #1
Tygra
39
4
Homework Statement
Solving a Continuous Beam
Relevant Equations
Fourth Order Differential Equation (FDM)
Hi all,

I am learning how to solve differential equtions using the finite diference method. In particular, for beams under a uniformly distributed load. For a simply supported beam this is quite easy. The boundary conditions are that at each end the displacement equal zero, and using the fourth order finite difference equation you construct the matrix. The FDM for the fourth order differential equation is:

1726009001269.png


For a 10m beam using 12 points, and applying the boundary conditions, the matrix looks like:

6 -4 1 0 0 0 0 0 0 0
-4 6 -4 1 0 0 0 0 0 0
1 -4 6 -4 1 0 0 0 0 0
0 1 -4 6 -4 1 0 0 0 0
0 0 1 -4 6 -4 1 0 0 0
0 0 0 1 -4 6 -4 1 0 0
0 0 0 0 1 -4 6 -4 1 0
0 0 0 0 0 1 -4 6 -4 1
0 0 0 0 0 0 1 -4 6 -4
0 0 0 0 0 0 0 1 -4 6

The fourth order differential is equal to the uniformly distributed load q(x). So the right-hand side is:

1726007969705.png


So, I have solved the displacements for a simply supported beam quite easily, but now I want to go further and use the FDM to solve for a continuous beam that say has 3 supports - pin support at one end with a roller support in the middle of the beam and at the other end.

yvCEb.png


There are 3 boundary conditions: The displacement is zero at each support. My question is: how do I construct the matrix using the FDM for this problem? I believe it would be called a multi-point boundary value problem. Everything I have searched only considers a two-point boundary value problem, like for the simply supported beam. I am wondering if anyone on here has knowledge on how to use the FDM for a boundary value problem for more than two points?

Many thanks in advance!

Tygra
 

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  • #2
Firstly, use an odd number of nodes so one of them is exactly at the center point.

Then do exactly the same as at any other node where the displacement is known: you don't need to solve for that value, so you don't need an equation for it in your system. Wherever it occurs in the equations for the other nodes you substitute its known value and shift those terms to the right hand side.

Alternatively, if the value of node is known, you can replace row of the matrix with and replace row of the right hand side with the known value of node .
 
  • #3
Hi pasmith, thank you for responding.

So if I use 13 nodes, node 7 will be the centre, and thus I eliminate the column and row with respect to node 7? Which is what I did for the first and the last nodes where the displacement equals zero. So I am left with a matrix of 10 x 10 shown below:
Code:
     6    -4     1     0     0     0     0     0     0     0
    -4     6    -4     1     0     0     0     0     0     0
     1    -4     6    -4     0     0     0     0     0     0
     0     1    -4     6     1     0     0     0     0     0
     0     0     0     1     6    -4     1     0     0     0
     0     0     0     0    -4     6    -4     1     0     0
     0     0     0     0     1    -4     6    -4     1     0
     0     0     0     0     0     1    -4     6    -4     1
     0     0     0     0     0     0     1    -4     6    -4
     0     0     0     0     0     0     0     1    -4     6

I regards to moving things to the right-hand side I am not totally clear what you mean. At the moment my right-hand side looks like this:

1726067889511.png

Do you mean I have to pull a 6, -4 or 1 to the right-hand side?
 
  • #4
Can anyone help me on this?
 

FAQ: Multi-point Boundary Value Problems Using Finite Difference Method

What are multi-point boundary value problems (MPBVPs)?

Multi-point boundary value problems are differential equations that require the solution to satisfy conditions at multiple points rather than just at the endpoints of the interval. These problems often arise in various fields such as engineering and physics, where the behavior of a system is influenced by conditions at several points along the domain.

How does the finite difference method work for solving MPBVPs?

The finite difference method approximates the derivatives in the differential equations using difference quotients. By discretizing the domain into a grid, the differential equations are transformed into a system of algebraic equations. The boundary conditions at multiple points are incorporated into this system, allowing for the numerical solution of the MPBVP.

What are the advantages of using the finite difference method for MPBVPs?

The finite difference method is relatively simple to implement and understand. It provides a straightforward way to approximate solutions to differential equations, and it can handle complex boundary conditions. Additionally, it is versatile and can be applied to a wide range of problems, making it a popular choice among researchers and engineers.

What are some common challenges when applying the finite difference method to MPBVPs?

Some common challenges include ensuring stability and convergence of the numerical solution, especially for stiff equations. Additionally, accurately implementing boundary conditions at multiple points can be complex, and the choice of grid size can significantly affect the accuracy of the solution. Careful consideration of these factors is essential for effective application.

Are there alternative methods to the finite difference method for solving MPBVPs?

Yes, there are several alternative methods, including the shooting method, finite element method, and collocation method. Each of these methods has its own strengths and weaknesses, and the choice of method often depends on the specific characteristics of the problem being solved, such as the type of differential equation and the boundary conditions involved.

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