Numerical recovery of the initial condition in the Cauchy problems for linear parabolic and hyperbolic equations

Background. The theory of solving inverse problems of mathematical physics is one of the most actively developing branches of modern mathematics. The interest of researchers in such problems is primarily due to the large number of their applications that have appeared in recent years in connection with the rapid development of physics and technology. Despite the large number of methods for solving inverse problems, at present, there is still a great need for the further development of new methods of solving that take into account the incorrectness of a number of inverse problems. In this paper, we propose numerical methods for solving one class of inverse problems, namely, problems of recovering the initial conditions for equations of parabolic and hyperbolic types.
Materials and methods. The technique for constructing numerical methods for solving problems of recovering initial conditions for linear parabolic and hyperbolic equations is as follows. According to the well-known formulas for the generalized solution of linear parabolic and hyperbolic equations, a transition is made to the equivalent initial problems of linear integral equations of the first kind, which are then solved approximately using the continuous operator method. For this, an auxiliary system of linear differential equations is compiled and solved, which is then solved by the numerical Euler method. At the same time, numerical examples show that due to a suitable number of steps of the Euler method, a regularization of the solution of the problem can be achieved (if necessary). The convergence of the method is substantiated in terms of the stability theory of the solution of differential equations.
Results. Numerical methods are developed for the approximate solution of the problem of recovering the initial condition for linear parabolic and hyperbolic equations. The authors have successfully applied the continuous operator method to the solution of the above problem. The solution of a number of model examples showed the effectiveness of the proposed results.
Conclusions. Effective numerical methods are proposed for solving one class of inverse problems of mathematical physics, namely, the problem of recovering the initial condition in Cauchy problems for linear equations of hyperbolic and parabolic types. Numerical examples show that the continuous operator method can be successfully applied to the solution of the indicated types of inverse problems of mathematical physics.