‎in this paper, we discuss about existence of solution forintegro-differential system and then we solve it  by using the petrov-galerkin method. in the petrov-galerkin method choosing the trial and test space is important, so  we use alpert multi-wavelet as basisfunctions for these spaces. orthonormality is one of theproperties of alpert multi-wavelet which helps us to reducecomputations in the...

Time-stepping methods that guarantee to avoid spurious fixed points are said to be regular. For fixed stepsize Runge-Kutta formulas, this concept has been well studied. Here, the theory of regularity is extended to the case of embedded Runge-Kutta pairs used in variable stepsize mode with local error control. First, the limiting case of a zero error tolerance is considered. A recursive regulari...