Boundary value problems for systems of equations of mixed and composite types

The theory of general elliptic boundary value problems for systems of equations is explained in Chapter 1 (see also [R1, Ch.X], and bibliography there).

This volume deals with first and second order complex equations of hyperbolic and mixed types. Various general boundary value problems for linear and quasilinear complex equations are investigated in detail. To obtain results for complex equations of mixed types, some discontinuous boundary value problems for elliptic complex equations are discusse

This article examines the unique solvability of a boundary value problem for hyperbolic systems under periodic conditions. To solve the stated problem, the Dzhumabaev method is employed, which transforms the original boundary value problem into an equivalent multipoint boundary value problem for ordinary differential equations with parameters, incorporating both initial and additional boundary conditions. Furthermore, the study establishes a connection between the unique solvability of the original problem and the invertibility of the matrix . Recurrent relations for determining the elements of the inverse matrix are also derived.

In this paper we consider a boundary value problem for systems of Fredholm type integral-differential equations with involutive transformation, containing derivative of the required function on the right-hand side under the integral sign. Applying properties of an involutive transformation, original boundary value problem is reduced to a boundary value problem for systems of integral-differential equations, containing derivative of the required function on the right side under the integral sign. Assuming existence of resolvent of the integral equation with respect to the kernel K˜2(t, s) (this is the kernel of the integral equation that contains the derivative of the desired function) and using properties of the resolvent, integral-differential equation with a derivative on the right-hand side is reduced to a Fredholm type integral-differential equation, in which there is no derivative of the desired function on the right side of the equation. Further, the obtained boundary value problem is solved by the parametrization method created by Professor D. Dzhumabaev. Based on this method, the problem is reduced to solving a special Cauchy problem with respect to the introduced new functions and to solving systems of linear algebraic equations with respect to the introduced parameters. An algorithm to find a solution is proposed. As is known, in contrast to the Cauchy problem for ordinary differential equations, the special Cauchy problem for systems of integral-differential equations is not always solvable. Necessary conditions for unique solvability of the special Cauchy problem were established. By using results obtained by Professor D. Dzhumabaev, necessary and sufficient conditions for the unique solvability of the original problem were established.

On solvability and unique solvability of two-point singular boundary value problems

In this paper, oblique derivative boundary value problems for second order equations of mixed (elliptic-hyperbolic) type in multiply connected domains is discussed. Firstly the representation of solutions forthe above boundary value problem is given, afterwards the uniqueness and existence of solutions of the above problem are stated. In book [1], the author proposed the Dirichlet boundary value problem (Tricomi problem) forsecond order equations of mixed type in multiply connected domains. In [2, 3], the author only discussed the Dirichlet problem (Tricomi T 2) for the equation u xx +sgny u yy = 0 in a special doubly connected domain. Up to now we have not seen that other authors have solved it in multiply connected domains. In the present paper, we try to discuss the oblique derivative problem for second order equations of mixed type in multiply connected domains, which includes the Dirichlet problem (Problem T 2) as a special case.Key wordsoblique derivative problemsequations of mixed typemultiply connected domains.

This article is devoted to study the boundary value problems of the first and second kind with respect to the spatial variable for a mixed inhomogeneous differential equation of parabolic-hyperbolic type with a fractional Caputo operator in a rectangular domain. In the study of such boundary value problems, we abandoned the boundary value condition with respect to the first argument and instead it is used additional gluing condition. In this case, in the justification of the unique solvability of the problems, the conditions on the boundary domain are removed. This allowed us to weaken the criterion for the unique solvability of boundary value problems under consideration. The solution is constructed in the form of Fourier series with eigenfunctions corresponding to homogeneous spectral problems. Estimates for the convergence of Fourier series are obtained as a regular solution of this mixed equation.

Sufficient conditions are established for the solvability of the boundary value problem where p : C(I; Rn ) × C(I; Rn ) → L(I; Rn ), q : C(I; Rn ) → L(I; Rn ), l : C(I; Rn ) × C(I; Rn ) → Rn , and cn : C(I; Rn ) → Rn are continuous operators, and p(x, ·) and l(x, ·) are linear operators for any fixed x ∈ C(I; Rn ).

We present our recent results on the mathematical analysis of shock diffraction by two-dimensional convex cornered wedges in compressible fluid flow governed by the potential flow equation. The shock diffraction problem can be formulated as an initial-boundary value problem, which is invariant under self-similar scaling. Then, by employing its self-similar invariance, the problem is reduced to a boundary value problem for a first-order nonlinear system of partial differential equations of mixed elliptic-hyperbolic type in an unbounded domain. It is further reformulated as a free boundary problem for a nonlinear degenerate elliptic system of first-order in a bounded domain with a boundary corner whose angle is bigger than π. A first global theory of existence and regularity has been established for this shock diffraction problem for the potential flow equation.

Integral equations of mixed Voltera-Fredholm type arise in various physical and biological problems. In the present paper, we obtain the approximate solution of the nonlinear Volterra-Hammerstein integral equations of mixed type in terms of Taylor polynomial. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments.

The simulation of sedimentary basins aims at reconstructing its historical evolution in order to provide quantitative predictions about phenomena leading to hydrocarbon accumulations. The kernel of this simulation is the numerical solution of a complex system of non-linear partial differential equations (PDE) of mixed parabolic-hyperbolic type in 3D. A discretisation and linearisation of this system leads to very large, ill-conditioned, non-symmetric systems of linear equations with three unknowns per mesh cell, i.e. pressure, geostatic load, and oil saturation.

This article studies the Keldysh problem for a three-dimensional equation of mixed type with three singular coefficients in a semi-infinite parallelepiped. Based on the completeness property of eigenfunction systems of two one-dimensional spectral problems, the uniqueness theorem is proved. To prove the existence of a solution to the problem, the Fourier spectral method based on the separation of variables is used. The solution to this problem is constructed in the form of a sum of a double Fourier-Bessel series. In substantiating the uniform convergence of the constructed series, we used asymptotic estimates of the Bessel functions of the real and imaginary argument. Based on them, estimates were obtained for each member of the series, which made it possible to prove the convergence of the series and its derivatives to the second order inclusive, as well as the existence theorem in the class of regular solutions.

Introduction. This study investigates the possibility of increasing the accuracy of numerically solving boundary value problems using the modified Bubnov-Galerkin method with a linear ordinary differential equation, where the coefficients and the right-hand side are continuous functions. The order of the differential equation n must be less than the number of coordinate functions m.Materials and Methods. A modified Petrov-Galerkin method was used to numerically solve the boundary value problem. It employs a system of linearly independent power-type basis functions on the interval [−1,1], each normalized by the unit Chebyshev norm. The system of linear algebraic equations includes only the linearly independent boundary conditions of the original problem.Results. For the first time, an integral quadrature formula with a 22nd order error was developed for a uniform grid. This formula is used to calculate the matrix elements and coefficients in the right-hand side of the system of linear algebraic equations, taking into account the scalar product of two functions based on the new quadrature formula. The study proves a theorem on the existence and uniqueness of a solution for boundary value problems with general non-separated conditions, provided that n linearly independent particular solutions of a homogeneous differential equation of order n are known.Discussion and Conclusion. The hydrodynamic problem in a viscous strong boundary layer with a third-order equation was precisely solved. The analytical solution was compared with its numerical counterpart, and the uniform norm of their difference did not exceed 5·10‒15. The formulas derived using the generalized Bubnov-Galerkin method may be useful for solving boundary value problems with linear ordinary differential equations of the third and higher orders.

The work is devoted to the study of the solvability of the inverse boundary value problem with an unknown time depended coefficient for a sixth-order Boussinesq equation with an additional integral condition of the first kind is investigated. First, the given problem is reduced to an equivalent problem in a certain sense. Then, using the Fourier method the equivalent problem is reduced to solving the system of integral equations. The existence and uniqueness of a solution to the system of integral equation is proved by the contraction mapping principle. This solution is also the unique solution to the equivalent problem. Finally, by equivalence, the theorem of existence and uniqueness of a classical solution to the given problem is proved. The sixth order Boussinesq-Type problem is discretized using the Quintic B-spline (QB-spline) collocation technique and reshaped as non-linear least-squares optimization of the Tikhonov regularization function. This is numerically solved by means of the MATLAB subroutine lsqnonlin tool. Both perturbed data and analytical solution are inverted. Numerical outcomes are reported and discussed. In addition, the von Neumann stability analysis for the proposed numerical approach has also been discussed.

Sufficient conditions guaranteeing the unique existence of a bounded solution of quasilinear boundary value problems are established and a successive approximation method for constructing the solution is described, together with the proof of the stability of the used method.