To solve the first-order differential equation derived from the problem of a free-falling object and the problem arising from Newton’s law of cooling, the study compares the numerical solutions obtained from Picard’...To solve the first-order differential equation derived from the problem of a free-falling object and the problem arising from Newton’s law of cooling, the study compares the numerical solutions obtained from Picard’s and Taylor’s series methods. We have carried out a descriptive analysis using the MATLAB software. Picard’s and Taylor’s techniques for deriving numerical solutions are both strong mathematical instruments that behave similarly. All first-order differential equations in standard form that have a constant function on the right-hand side share this similarity. As a result, we can conclude that Taylor’s approach is simpler to use, more effective, and more accurate. We will contrast Rung Kutta and Taylor’s methods in more detail in the following section.展开更多
Using the Picard iteration method and treating the involved integration by numerical quadrature formulas, we propose a numerical scheme for the second kind nonlinear Volterra integral equations. For enlarging the conv...Using the Picard iteration method and treating the involved integration by numerical quadrature formulas, we propose a numerical scheme for the second kind nonlinear Volterra integral equations. For enlarging the convergence region of the Picard iteration method, multistage algorithm is devised. We also introduce an algorithm for problems with some singularities at the limits of integration including fractional integral equations. Numerical tests verify the validity of the proposed schemes.展开更多
This paper deals with the problem of finding solutions to the Picard boundary problem. In our approacn, by means of the homotopy method, the equation considered is linked to a simpler equation by introducing a paramet...This paper deals with the problem of finding solutions to the Picard boundary problem. In our approacn, by means of the homotopy method, the equation considered is linked to a simpler equation by introducing a parameter. We first find the solutions of the simpler equation, and give a priori estimates of the equation we considered, and then one can obtain the solutions of Picard boundary problem by following the path of solutions of Cauchy problem.展开更多
This paper is devoted to the study of approximation of the solution for the differential equation whose coefficients are almost period functions. To this end the authors establish the estimation of the solution of gen...This paper is devoted to the study of approximation of the solution for the differential equation whose coefficients are almost period functions. To this end the authors establish the estimation of the solution of general linear differential equation for infinite interval case. For finite interval case, this equation was investigated by G. Tamarkin([1]) applying the Picard method of successive approximation.展开更多
文摘To solve the first-order differential equation derived from the problem of a free-falling object and the problem arising from Newton’s law of cooling, the study compares the numerical solutions obtained from Picard’s and Taylor’s series methods. We have carried out a descriptive analysis using the MATLAB software. Picard’s and Taylor’s techniques for deriving numerical solutions are both strong mathematical instruments that behave similarly. All first-order differential equations in standard form that have a constant function on the right-hand side share this similarity. As a result, we can conclude that Taylor’s approach is simpler to use, more effective, and more accurate. We will contrast Rung Kutta and Taylor’s methods in more detail in the following section.
文摘Using the Picard iteration method and treating the involved integration by numerical quadrature formulas, we propose a numerical scheme for the second kind nonlinear Volterra integral equations. For enlarging the convergence region of the Picard iteration method, multistage algorithm is devised. We also introduce an algorithm for problems with some singularities at the limits of integration including fractional integral equations. Numerical tests verify the validity of the proposed schemes.
文摘This paper deals with the problem of finding solutions to the Picard boundary problem. In our approacn, by means of the homotopy method, the equation considered is linked to a simpler equation by introducing a parameter. We first find the solutions of the simpler equation, and give a priori estimates of the equation we considered, and then one can obtain the solutions of Picard boundary problem by following the path of solutions of Cauchy problem.
文摘This paper is devoted to the study of approximation of the solution for the differential equation whose coefficients are almost period functions. To this end the authors establish the estimation of the solution of general linear differential equation for infinite interval case. For finite interval case, this equation was investigated by G. Tamarkin([1]) applying the Picard method of successive approximation.
