Solving Diffuse Light Simulation with Double Integral Equation?

In summary, the conversation discusses a program being developed to simulate diffuse light and the need to solve a specific equation involving W, T, and L. The speaker is new to differential and integral equations and is seeking help. A possible solution is suggested involving discretizing the equation and using linear algebra solvers like BLAS or LAPACK. This method can handle large matrix dimensions.
  • #1
Luonnos
2
0
Recently, I've been working on a program to simulate diffuse light, and I've hit a snag. I need to solve (at least so that a computer can compute L(x) quickly) something of the form:

[tex]
L(x)=T(x)+c\int_0^{l_2}\int_0^{l_1} W(x,u_1,u_2) L(u_1) du_1 du_2
[/tex]

W and T are pretty well behaved, and I can compute them and their integrals in a small amount of time. I'm pretty new to differential and integral equations, so any help would be appreciated.
 
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  • #2
If you discretize this on a computer, then L will be a column vector with dimension equal to the number of grid points. First, you should be able to integrate out u2, since L is a function only of u1. Then I think you can write your equation as follows, where L and T are column vectors, c is a constant, I is the identity matrix, and W is a square matrix:
[tex]L = T + c W L[/tex]
[tex](I-c W) L = T[/tex]
[tex]L = (I-c W)^{-1} T [/tex]
Then you can use a linear algebra solver like BLAS or LAPACK to solve for L. These will find solutions even if the matrix dimensions are on the order of 1000's x 1000's.
 

Related to Solving Diffuse Light Simulation with Double Integral Equation?

1. What is a double integral equation?

A double integral equation is a mathematical expression that involves the integration of a function of two variables over a two-dimensional region. It is written in the form of ∬f(x,y)dA, where f(x,y) represents the function being integrated and dA represents the differential area element.

2. How is a double integral equation different from a single integral equation?

A single integral equation involves the integration of a function of one variable over a one-dimensional region, while a double integral equation involves the integration of a function of two variables over a two-dimensional region. This means that a double integral equation calculates the volume under a surface, while a single integral equation calculates the area under a curve.

3. What are some common applications of double integral equations?

Double integral equations have many real-world applications, including calculating areas and volumes, finding the center of mass of an object, determining the average value of a function over a region, and solving problems in physics and engineering, such as calculating moments of inertia and work done by a variable force.

4. How do you solve a double integral equation?

Solving a double integral equation involves finding the antiderivative of the function being integrated and then evaluating it at the limits of integration. This can be done using various integration techniques, such as u-substitution, integration by parts, or partial fraction decomposition. It is also important to correctly set up the limits of integration based on the given region.

5. What are some common mistakes to avoid when working with double integral equations?

Some common mistakes to avoid when working with double integral equations include forgetting to correctly set up the limits of integration, using the wrong integration technique, and not simplifying the integrand before integrating. It is also important to check for symmetry in the region of integration, as this can often simplify the calculation. Additionally, always double check your calculations and be aware of common integration mistakes, such as forgetting to include the constant of integration.

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