How to solve a system of integral equation

Question

I am stuck on the following problem: given this set of integral equations: $$\int_{\lambda_0}^{\lambda_1}d\lambda S(\lambda)W(\lambda,T_0)=k_1\int_{\lambda_0}^{\lambda_1}d\lambda S(\lambda)W(\lambda,T_1)$$ $$\int_{\lambda_0}^{\lambda_1}d\lambda S(\lambda)W(\lambda,T_0)=k_2\int_{\lambda_0}^{\lambda_1}d\lambda S(\lambda)W(\lambda,T_2)$$ $$...........................$$ $$\int_{\lambda_0}^{\lambda_1}d\lambda S(\lambda)W(\lambda,T_0)=k_N\int_{\lambda_0}^{\lambda_1}d\lambda S(\lambda)W(\lambda,T_N)$$ where $W(\lambda,T)$ is a known function, I need to find $S(\lambda)$. Putting $\displaystyle S(\lambda)=\sum_{j=0}^Na_j\lambda^j$ can the system of integral equations can be reduced to an algebraic equation? Thanks in advance.

Answer 1

Let $w_i(\lambda):=W(\lambda, T_0)-k_iW(\lambda,T_i)$, and $<f,g>:=\int_{\lambda_0}^{\lambda_1}d\lambda f(\lambda)g(\lambda)$, which are scalars.

Then take any $F(\lambda)$ such that the integrals below are defined, and let

$$S(\lambda):=F(\lambda)+\sum_{i=1}^N a_iw_i(\lambda).$$

The given equations imply

$$<S,w_j>=<F,w_j>+\sum_{i=1}^N a_i<w_i,w_j>=0.$$

Unless the system is singular, you can solve for the $a_i$ and obtain $S$.