Non-linear problems with non-monotonic blow-up solutions: Non-local transformations, test problems, exact solutions, and numerical integration

The method of non-local transformations is proposed for numerical integration of non-linear Cauchy problems having non-monotonic blow-up solutions. In such problems there exists a singular point whose position is unknown in advance (for this reason, the standard numerical methods for solving blow-up problems can lead to significant errors). In addition, the non-monotonic behavior of the solution excludes a possibility of applying the hodograph transformation and some other methods that are used for numerical investigation of simpler problems having monotone blow-up solutions. In this paper, the method is described for numerical integration of similar problems for non-linear n th-order ordinary differential equations xt(n)=g(t,x,xt′,…,xt(n−1)), based on the introduction of a new non-local independent variable ξ, which is related to the original variables t and x by the equation ξt′=g(t,x,xt′,…,xt(n−1),ξ), and the subsequent transformation of the original problem to the Cauchy problem for the corresponding system of first-order differential equations. With a suitable choice of the regularizing function g, the proposed method leads to problems whose solutions are presented in parametric form and do not have blowing-up singular points; therefore the transformed problems allow the application of the standard fixed-step numerical methods. A number of test problems with non-monotonic blow-up is constructed for differential equations of the first, second, third, and fourth orders, which have exact solutions expressed in elementary functions. The numerical integration of test problems has shown high efficiency of methods based on non-local transformations of a special type (and the practical inapplicability of the arc-length transformation for numerical solving blow-up problems with ODEs of high order). In addition to problems for single ODEs, Cauchy problems for systems of coupled equations also are considered. Some recommendations are given on a suitable choice of a variable step in direct adaptive numerical methods in which the transformations of equations are not used. It is important to note that the method of non-local transformations can also be used to numerically integrate other problems with large solution gradients (including problems of boundary-layer type).