On the mixed problem for quasilinear pseudoparabolic equation

Principles of Construction of Adjoint Operators in Non-Linear Problems Dual Spaces and Adjoint Operators Construction of Adjoint Operators Based on Using the Lagrange Identity Definition of Adjoint Operators Based on Using Taylor's Formula Operators of the Class D and their Adjoint Operators Properties of Adjoint Operators Constructed on the Basis of Various Principles General Properties of Main and Adjoint Operators Corresponding to Non-Linear Operators Properties of Operators of the Class D Properties of Adjoint Operators Constructed with the Use of the Taylor Formula Solvability of Main and Adjoint Equations in Non-Linear Problems Main and Adjoint Equations. Problems Solvability of the Equation F(u) = y Solvability of the Equation A(u)v = y Solvability of the Equation A(u)v = y Solvability of the Equation A*(u)w = p Solvability of the Equation A*(u)w = p Transformation Groups, Conservation Laws and Construction of the Adjoint Operators in Non-Linear Problems Definitions. Non-Linear Equations and Operators. Conservation Laws Transformation of Equations Adjoint Equations Relationship between Different Adjoint Operators General Remarks on Constructing the Adjoint Equations with the Use of the Lie Groups and Conservation Laws Construction of Adjoint Operators with Prescribed Properties The Noether Theorem, Conservation Laws and Adjoint Operators On Some Applications of Adjoint Equations Perturbation Algorithms in Non-Linear Problems Perturbation Algorithms for Original Non-Linear Equations and Equations Involving Adjoint Operators Perturbation Algorithms for Non-Linear Functionals Based on Using Main and Adjoint Equations Spectral Method in Perturbation Algorithms Justification of the N-th Order Perturbation Algorithms Convergence Rate Estimates for Perturbation Algorithms. Comparison with the Successive Approximation Method Justification of Perturbation Algorithms in Quasi-Linear Elliptic Problems Adjoint Equations and the N-th Order Perturbation Algorithms in Non-Linear Problems of Transport Theory Some Problems of Transport Theory The N-th Order Perturbation Algorithms for an Eigenvalue Problem A Problem of Control and its Approximate Solution with the Use of Perturbation Algorithms Investigation and Approximate Solution of a Non-Linear Problem for the Transport Equation Adjoint Equations and Perturbation Algorithms for a Quasilinear Equation of Motion Statement of the Problem. Basic Assumptions. Operator Formulation Transformation of the Problem. Properties of the Non-Linear Operator Adjoint Equation An Algorithm for Computing the Functional The Problem on Chemical Exchange Processes Adjoint Equations and Perturbation Algorithms for a Non-Linear Mathematical Model of Mass Transfer in Soil Mathematical Models of Mass Transfer in Soil Formulation of a Non-Linear Mathematical Model Transformation of the Problem. Properties of the Non-Linear Operator Perturbation Algorithm. Adjoint Equation Approximate Solution of the Problem on Finding an Effective Dispersion Coefficient An Algorithm for Solving the Problem Applications of Adjoint Equations in Science and Technology Adjoint Equations in Data Assimilation Problems Application of Adjoint Equations for Solving the Problem of Liquid Boundary Conditions in Hydrodynamics Shape Optimization Using Adjoint Equation Approaches Global Transport of Pollutants Problems of Climate Change Sensitivity in Various Regions of the World

The integral-differential equations describe many phenomena in different field of mechanical and nuclear engineering, chemistry, astronomy, biology and potential theory. The resolution of boundary problems for these equations are the subject of several papers in which the authors have approached in most cases numerical methods: Monte Carlo, truncated series of Chebyshev polynomials, the fictitious domain method, S N method and the finite element method (1-4). An exact solution was found only in the particular cases. Generally, these are obtained with the help of the methods of mathematical analysis, abstract functional analysis and the spectral methods (5). In recent years, the basic ideas of the homotopy, which is a concept of the topology and differential geometry, were used to obtain the approximate solutions for a wide class of differential, integral and integral-differential equations. We mention here the homotopy perturbation method (HPM) proposed by Ji-Huan He in 1998 and the homotopy analysis method (HAM) proposed by Liao in 1992. The perturbation methods approximate the solution of given problem by a series of small parameters. Unfortunately, the majority of non-linear problems have no small parameters and an unsuitable choice of these parameters can lead to bad effects. The new homotopy perturbation technique (HPM) embeds a parameter p that ranges from zero to one (6,7). When the embedding parameter is zero, we get a linear equation and if it is equal to one, we get the original transport equation. This embedding parameter that belongs to the interval (0, 1) can be considered as a small parameter. The homotopy analysis method (HAM) is one of the most effective methods to obtain the exact and approximate series solutions for the integral-differential equations (8). Being a great freedom in choosing of the initial guess functions and the auxiliary functions, are proposed for these such forms that correspond to the conditions imposed at the ends of the spatial interval and thus the algorithm becomes fast convergent. In the papers (9-11) is shown that the homotopy perturbation method, variational iteration method and Adomian decomposition method are only special cases of the homotopy analysis method solutions. Today, a new homotopy method is added to the above. With this we can solve the multipoint boundary problem for a stationary transport equation in the non-homogeneous media. Comparative studies between the solutions obtained with the help of mentioned algorithms for a numerical example with the appropriate exact solutions illustrate the accuracy and computational efficiency of these new methods.

We describe continuous linear operators acting in a countable inductive limit of weighted Frchet spaces of entire functions of several complex variables and commuting in these spaces with systems of partial differentiation and translation operators. Under the made assumptions, the commutants of the systems of differentiation and translation operators coincide. They consist of convolution operators defined by an arbitrary continuous linear functional on . At that, we do not assume that the set of the polynomials is dense in . In the space topological dual to , we introduce the natural multiplication. Under this multiplication, the algebra is isomorphic to the aforementioned commutant with the usual multiplication, which is the composition of the operators. This isomorphism is also topological if is equipped by the weak topology, while the commutant is equipped by the weak operator topology. This implies that the set of the polynomials of the differentiation operators is dense in the commutant with topology of pointwise convergence. We also study the possibility of representing an operator in the commutant as an infinite order differential operator with constant coefficients. We prove the immediate continuity of linear operators commuting with all differentiation operators in a weighted (LF)-space of entire functions isomorphic via Fourier-Laplace transform to the space of infinitely differentiable functions compactly supported in a real multi-dimensional space.

Weak solutions of impulsive pseudoparabolic equations with an infinitesimal transition layer

An inverse dynamics method is introduced for constructing exact special-case solutions for hybrid coordinate ordinary/partial systems of differential equations (hybrid ODE/PDE systems). The solution is constructed such that it lies near a given approximate numerical solution, and therefore the special-case solutions can be generated in a versatile and physically meaningful fashion and can serve as a benchmark problem to validate approximate solution methods. The exact solution is constructed such that it is a differentiable, continuous-function neighbor of the given approximate numerical solution. This continuous solution is then substituted into the governing system of ODEs/PDEs and a full complement of distributed and boundary forces is determined algebraically to exactly satisfy the differential equations. This process has been automated by computer symbol manipulation. Since the exact special-case algebraic solutions can be evaluated anywhere in space and time, this approach is ideally suited to providing a true exact motion and the corresponding forces for studying the convergence errors in a family of approximate solutions. This methodology makes it possible for one to rigorously determine exact solution errors for a significant class of ODE/PDE systems for which the initial-value problem is not, in general, exactly solvable. We explore the utility of this method in validating numerical solution methods.

Adjoint operators for the natural discretizations of the divergence, gradient and curl on logically rectangular grids

We consider boundary value problems where the operator defined in a domain and the boundary operators depend on a small parameter. Elliptic and properly elliptic problems with a small parameter are defined. It is proved that small parameter ellipticity is a necessary and sufficient condition for the existence of

Real-Time Point Process Filter for Multidimensional Decoding Problems Using Mixture Models

Approximate analytical solutions of singularly perturbed fourth order boundary value problems using differential transform method

In this paper, an approximate analytical solution for Frank-Kamenetskii equation modeling a thermal explosion in a sphere, is obtained. The approximate solution is obtained by perturbation methods in terms of small and large distance parameter. The approximate solution is compared with the numerical solution obtained from an initial value problem formulation to the original boundary value problem. The approximate solution obtained is valid for all values of the distance parameter. For the original boundary value problem and for a given Frank-Kamenetskii parameter, a nonlinear algebraic equation needs to be solved to be able to apply the approximate solution.

The Whitham equation is a non-local model for nonlinear dispersive water waves. Since this equation is both nonlinear and non-local, exact or analytical solutions are rare except for in a few special cases. As such, an analytical method which results in minimal error is highly desirable for general forms of the Whitham equation. We obtain approximate analytical solutions to the non-local Whitham equation for general initial data by way of the optimal homotopy analysis method, through the use of a partial differential auxiliary linear operator. A method to control the residual error of these approximate solutions, through the use of the embedded convergence control parameter, is discussed in the context of optimal homotopy analysis. We obtain residual error minimizing solutions, using relatively few terms in the solution series, in the case of several different kernels and associated initial data. Interestingly, we find that for a specific class of initial data, there exists an exact solution given by the first term in the homotopy expansion. A specific example of initial data which satisfies the condition producing an exact solution is included. These results demonstrate the applicability of optimal homotopy analysis to equations which are simultaneously nonlinear and non-local.

In this paper, we consider the question on study of zeros of an entire function of one class, which coincides with quasi-polynomials of exponential type. Eigenvalue problems for some classes of differential operators on a segment are reduced to a similar problem. In particular, the studied problem is led by the eigenvalue problem for a linear differential equation of the third order with regular boundary value conditions in the space W^3_2(0, 1). The studied entire function is adequately characteristic determinant of the spectral problem for a third-order linear differential operator with periodic boundary value conditions. An algorithm to construct a conjugate indicator diagram of an entire function of one class is indicated, which coincides with exponential type quasi-polynomials with comparable exponents according to the monograph by A.F. Leontyev. Existence of a countable number of zeros of the studied entire function in each series is proved, which are simultaneously eigenvalues of the above-mentioned third-order differential operator with regular boundary value conditions. We determine distance between adjacent zeros of each series, which lies on the rays perpendicular to sides of the conjugate indicator diagram, that is a regular hexagon on the complex plane. In this case, zero is not an eigenvalue of the considered operator, that is, zero is a regular point of the operator. Fundamental difference of this work is finding the corresponding eigenfunctions of the operator. System of eigenfunctions of the operator corresponding in each series is found. Adjoint operator is constructed.

We discuss a problem of constructing self-adjoint ordinary differential operators starting from self-adjoint differential expressions based on the general theory of self-adjoint extensions of symmetric operators outlined in [1]. We describe one of the possible ways of constructing in terms of the closure of an initial symmetric operator associated with a given differential expression and deficient spaces. Particular attention is focused on the features peculiar to differential operators, among them on the notion of natural domain and the representation of asymmetry forms generated by adjoint operators in terms of boundary forms. Main assertions are illustrated in detail by simple examples of quantum-mechanical operators like the momentum or Hamiltonian.

In quantum field theory, the fractional Kundu-Eckhaus and massive Thirring models are nonlinear partial differential equations under fractional sense inside nonlinear Schrödinger class. In this study, approximate analytical solutions of such complex nonlinear fractional models are acquired by means of conformable residual power series method. This method presents a systematic procedure for constructing a set of periodic wave series solutions based on the generalization of conformable power series and gives the unknown coefficients in a simple pattern. By plotting the solutions behavior of the models; the convergence regions in which the solutions coincide to each other are checked for various fractional values. The approximate solutions generated by the proposed approach are compared with the exact solutions -if exist- and the approximate solutions obtained using qHATM and LADM. Numerical results show that the proposed method is easy to implement and very computationally attractive in solving several complex nonlinear fractional systems that occur in applied physics under a compatible fractional sense.

In this paper, classical and nonclassical symmetries of the potential KdV equation are considered. A catalogue of symmetry reductions for potential KdV equation is obtained using the classical Lie method and nonclassical method due to Bluman and Cole.[1] The Lie algebra consists of five finite parameter Lie group transformations; two being the scaling symmetry and the others being translations. By using the nonclassical method, four finite parameter group transformations are obtained. Two different types of solutions, approximate and exact solutions, are found by using the symmetries. Using the classical symmetries, only approximate series solutions are constructed, but using the nonclassical symmetries, both the exact solution and the approximate solution were found. The advantage of nonclassical symmetries is that the exact solution of the KdV equation can be found with it. Perturbation method is also used for approximate solutions.