We consider the inverse problem for identification of the coefficient in an elliptic partial differential equation inside of the unit square D, with overposed boundary data. This problem is not investigated enough in the literature due to lack of results about the uniqueness of the inverse problem. Following the main idea of the so-called Method of Variational Imbedding (MVI), we embed the solu...

Let Ω be an open, simply connected, and bounded region in R, d ≥ 2, and assume its boundary ∂Ω is smooth. Consider solving an elliptic partial differential equation Lu = f over Ω with zero Dirichlet boundary values. The problem is converted to an equivalent elliptic problem over the unit ball B, and then a spectral method is given that uses a special polynomial basis. With sufficiently smooth p...

In this paper Lie’s formalism is applied to deduce classes of solutions of a nonlinear partial differential equation (nPDE) of second order with quadratic nonlinearity. The equation has the meaning of a field equation appearing in the formulation of kinetic models. Similarity solutions and transformations are given in a most general form derived to the first time in terms of reciprocal Jacobian...

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This paper is concerned with the derivation and analysis of first-order necessary optimality conditions for a class multiobjective optimal control problems governed by an elliptic non-smooth semilinear partial differential equation. Using adjoint calculus inverse non-linear non-differentiable directional derivative solution map considered PDE, we extend concept strong stationarity to setting de...

In this paper we further improve the modified extended tanh-function method to obtain new exact solutions for nonlinear partial differential equations. Numerical applications of the proposed method are verified by solving the improved Boussinesq equation and the system of variant Boussinesq equations. The new exact solutions for these equations include Jacobi elliptic doubly periodic type, Weie...

In this paper we construct a kernel estimator of a time-varying coefficient of a strongly elliptic partial differential operator in a stochastic parablic equation. The equation is assumed diagonalizable, that is, all the operators have a common system of eigenfunctions. The meansquare convergence of the estimator is established. The rate of convergence is determined both by the smoothness of th...