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We establish the local (in time) solvability in the classical sense for the Cauchy problem and first and second boundary‐value problems on the half‐line for a nonlinear equation similar to Benjamin‐Bona‐Mahony‐Bürgers‐type equation. We also derive an a priori estimate that implies sufficient blow‐up conditions for the second boundary‐value problem. We obtain analytically an upper bound of the blow‐up time and refine it numerically using Richardson effective accuracy order technique. [ABSTRACT FROM AUTHOR]

This work develops the theory of the blow‐up phenomena for Joseph–Egri equation. The existence of the nonextendable solution of two initial‐boundary value problems (on a segment and a half‐line) is demonstrated. Sufficient conditions of the finite‐time blow‐up of these solutions, as well as the analytical estimates of the blow‐up time, are obtained. A numerical method that allows to precise the blow‐up moment for specified initial data is proposed. [ABSTRACT FROM AUTHOR]

In this paper, a new asymptotic-numerical approach to solving an inverse boundary value problem for a nonlinear singularly perturbed parabolic equation with time-periodic coefficients is proposed. An unknown boundary condition is reconstructed by using known additional information about the location of a moving front. An asymptotic analysis of the direct problem allows us to reduce the original inverse problem to that with a simpler numerical solution. Numerical examples demonstrate the efficiency of the method. [ABSTRACT FROM AUTHOR]

The work continues a series of articles devoted to the peculiarities of solving coefficient inverse problems for nonlinear singularly perturbed equations of the reaction-diffusion-advection-type with data on the position of the reaction front. In this paper, we place the emphasis on some problems of the numerical solving process. One of the approaches to solving inverse problems of the class under consideration is the use of methods of asymptotic analysis. These methods, under certain conditions, make it possible to construct the so-called reduced formulation of the inverse problem. Usually, a differential equation in this formulation has a lower dimension/order with respect to the differential equation, which is included in the full statement of the inverse problem. In this paper, we consider an example that leads to a reduced formulation of the problem, the solving of which is no less a time-consuming procedure in comparison with the numerical solving of the problem in the full statement. In particular, to obtain an approximate numerical solution, one has to use the methods of the numerical diagnostics of the solution's blow-up. Thus, it is demonstrated that the possibility of constructing a reduced formulation of the inverse problem does not guarantee its more efficient solving. Moreover, the possibility of constructing a reduced formulation of the problem does not guarantee the existence of an approximate solution that is qualitatively comparable to the true one. In previous works of the authors, it was shown that an acceptable approximate solution can be obtained only for sufficiently small values of the singular parameter included in the full statement of the problem. However, the question of how to proceed if the singular parameter is not small enough remains open. The work also gives an answer to this question. [ABSTRACT FROM AUTHOR]

The paper considers the features of numerical reconstruction of the advection coefficient when solving the coefficient inverse problem for a nonlinear singularly perturbed equation of the reaction-diffusion-advection type. Information on the position of a reaction front is used as data of the inverse problem. An important question arises: is it possible to obtain a mathematical connection between the unknown coefficient and the data of the inverse problem? The methods of asymptotic analysis of the direct problem help to solve this question. But the reduced statement of the inverse problem obtained by the methods of asymptotic analysis contains a nonlinear integral equation for the unknown coefficient. The features of its solution are discussed. Numerical experiments demonstrate the possibility of solving problems of such class using the proposed methods. [ABSTRACT FROM AUTHOR]