We study the Fredholm integro-differential equationby the wavelet method. Here (x) is the unknown function to be found, k(y) isa convolution kernel and g(x) is a given function. Following the idea in [7], theequation ...We study the Fredholm integro-differential equationby the wavelet method. Here (x) is the unknown function to be found, k(y) isa convolution kernel and g(x) is a given function. Following the idea in [7], theequation is discretized with respect to two different wavelet bases. We then havetwo different linear systems. One of them is a Toeplitz-Hankel system of the form(Hn + Tn)x = b where Tn is a Toeplitz matrix and Hn is a Hankel matrix. Theother one is a system (Bn+ Cn)y= d with condition number K = O(1) after adiagonal scaling. By using the preconditioned conjugate gradient (PCG) methodwith the fast wavelet transform (FWT) and the fast iterative Toeplitz solver, wecan solve the systems in O(nlog n) operations.展开更多
文摘We study the Fredholm integro-differential equationby the wavelet method. Here (x) is the unknown function to be found, k(y) isa convolution kernel and g(x) is a given function. Following the idea in [7], theequation is discretized with respect to two different wavelet bases. We then havetwo different linear systems. One of them is a Toeplitz-Hankel system of the form(Hn + Tn)x = b where Tn is a Toeplitz matrix and Hn is a Hankel matrix. Theother one is a system (Bn+ Cn)y= d with condition number K = O(1) after adiagonal scaling. By using the preconditioned conjugate gradient (PCG) methodwith the fast wavelet transform (FWT) and the fast iterative Toeplitz solver, wecan solve the systems in O(nlog n) operations.
