Sinc-collocation methods for weakly singular Fredholm integral equations of the second kind

AbstractIn this paper we propose new numerical methods for linear Fredholm integral equations of the second kind with weakly singular kernels. The methods are developed by means of the Sinc approximation with smoothing transformations, which is an effective technique against the singularities of the equations. Numerical examples show that the methods achieve exponential convergence, and in this sense the methods improve conventional results where only polynomial convergence have been reported so far

Error analyses of Sinc-collocation methods for exponential decay initial value problems

Nurmuhammad et al. developed Sinc-Nystr\"{o}m methods for initial value problems in which solutions exhibit exponential decay end behavior. In the methods, the Single-Exponential (SE) transformation or the Double-Exponential (DE) transformation is combined with the Sinc approximation. Hara and Okayama improved those transformations so that a better convergence rate could be attained, which was afterward supported by theoretical error analyses. However, due to a special function included in the basis functions, the methods have a drawback for computation. To address this issue, Okayama and Hara proposed Sinc-collocation methods, which do not include any special function in the basis functions. This study gives error analyses for the methods.Comment: Keywork: Ordinary differential equations, Initial value problems, Volterra integral equations, Sinc numerical methods, SE transformation, DE transformatio

Pseudocyst in the Pancreatic Tail Associated with Chronic Pancreatitis Successfully Treated by Transpapillary Cyst Drainage

We report a 50-year-old male with pseudocysts in the pancreatic tail associated with chronic pancreatitis successfully treated by transpapillary cyst drainage. He had previously undergone ultrasonography-guided percutaneous cyst drainage for a pancreatic pseudocyst in our hospital. He was readmitted due to abdominal pain and fever. Computed tomography showed recurrence of a pseudocyst in the pancreatic tail measuring 5 cm in diameter. Since conservative treatment failed, endoscopic retrograde pancreatography was performed. There was communication between the pseudocyst and the main pancreatic duct, and pancreatic duct stenosis proximal to the pseudocyst. First, transpapillary pancreatic duct drainage was performed using a plastic stent, but the pseudocyst did not decrease in size and became infected. After removal of the stent, a pigtail type nasocystic catheter was placed in the pseudocyst via the pancreatic duct. The pseudocyst infection immediately disappeared, and the pseudocyst gradually decreased and disappeared. After removal of the nasocystic catheter, no recurrence was observed. As transpapillary drainage of pancreatic pseudocyst, cyst drainage and pancreatic duct drainage have been reported. In our patient with pseudocyst in the pancreatic tail, duct drainage was ineffective and the pseudocyst was infected, whereas cyst drainage was very effective. We considered that cyst drainage by a nasocystic catheter was the first-line therapy as the transpapillary drainage of the pancreatic pseudocyst