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We study a class of nonlinear elliptic problems with Dirichlet conditions in the framework of the Sobolev anisotropic spaces with variable exponent, involving an anisotropic operator on an unbounded domain ${\varOmega }\subset \mathbb {R}^{N} (N \geq 2)$ . We prove the existence of entropy solutions avoiding sign condition and coercivity on the lower order terms.

For a partial differential equation of order 2n with constant coefficients in the domain G := {x = (x1,…, xm) : 0 < xj < 1 < ∞, j = 1,…, m, m ϵ ℕ} , we study the problem with conditions that are multipoint perturbations of the Dirichlet boundary conditions by using the Fourier method. To investigate the spectral properties of a multipoint problem, we use the operator of transformation R: L2 (G) → L2 (G) that establishes the relationship RL0 = LR between the self-adjoint operator L0 of the Dirichlet problem and the operator L of multipoint problem. The solution of the problem with homogeneous multipoint conditions is constructed in the form of Fourier series in the system of eigenfunctions of the operator of the problem. Moreover, the conditions for its existence and uniqueness are established.

The authors give sufficient conditions for the existence of at least three classical solutions to the nonlinear impulsive fractional boundary value problem with a p-Laplacian and Dirichlet conditions $$\begin{aligned} {\left\{ \begin{array}{ll} D^{\alpha }_{T^-}\varPhi _p(^cD^{\alpha }_{0^+}u(t))+|u(t)|^{p-2}u(t)= \lambda f(t,u(t)), &{}t\ne t_j,\ \ t\in (0,T),\\ \varDelta (D^{\alpha -1}_{T^-}\varPhi _p(^cD^{\alpha }_{0^+}u))(t_j)=I_j(u(t_j)),\\ u(0)=u(T)=0, \end{array}\right. } \end{aligned}$$ where $$\alpha \in (\frac{1}{p}, 1]$$ and $$p > 1$$ . Their approach is based on variational methods. The main result is illustrated with an example.

In this paper we present a pseudospectral method in the disk. Unlike the methods already known, the disk is not duplicated. Moreover, we solve the Laplace equation under nonhomogeneous Dirichlet, Neumann and Robin boundary conditions, as well as the biharmonic equation subject to nonhomogeneous Dirichlet conditions, by only using the elements of the corresponding differentiation matrices. It is worth mentioning that we do not use any quadrature, nor need to solve any decoupled system of ordinary differential equations, nor use any pole condition, nor require any lifting. We also solve several numerical examples to show the spectral convergence. The pseudospectral method developed in this paper is applied to estimate Sherwood numbers integrating the mass flux to the disk, and it can be implemented to solve Lotka–Volterra systems and nonlinear diffusion problems involving chemical reactions.

Na početku ovoga rada objašnjen je Fourierov red. Na nekoliko smo primjera prikazali razvoj funkcije u Fourierov red. U nastavku smo definirali parcijalne diferencijalne jednadžbe, njihove početne i rubne uvjete te vrste parcijalnih diferencijalnih jednadžbi. Tema ovog rada je diferencijalna Laplaceova jednadžba i električni potencijal. Upoznali smo se sa rubnim uvjetima kod Laplaceove jednadžbe te smo na primjeru pokazali kako se riješava Laplaceova parcijalna diferencijalna jednadžba u dvije dimenzije. Na kraju smo definirali električni potencijal. Primjer djelovanja odnosno korištenja Laplaceove jednadžbe u elektrostatici pokazali smo primjerom u kojem smo racčnali potencijal (raspodjelu naboja) između dvije paralelne ploče., At the beginning of this paper, the Fourier series was explained. We showed the development of a function into a Fourier series using several examples. In the following, we defined partial differential equations, their initial and boundary conditions, and types of partial differential equations. The topic of this paper is the differential Laplace equation and electric potential. We got acquainted with the boundary conditions of Laplace’s equation and we showed by example how to solve Laplace’s partial differential equation in two dimensions. Finally, we defined the electric potential. We showed an example of how the Laplace equation works in electrostatics with an example in which we calculated the potential (charge distribution) between two parallel plates.

The Stefan two-phase problem with an internal source, as well as the heat conduction problem for rectangular and curvilinear regions, has been solved by the method of rapid sine-expansion, which involves the Fourier series. The problem of heat conduction in a body of rectangular form was solved in a general form with the Dirichlet conditions and an internal heat source by the method of rapid expansion, and thereafter the problem for a body of curvilinear shape. The solutions obtained were used in considering the Stefan problem with unknown curvilinear interface. Expressions are presented for matching the input data of the problem, on satisfaction of which the solution will be universally continuous everywhere in the region of the rectangle together with derivatives up to the sixth order inclusive. The nonfulfillment of these conditions leads to the discontinuity of the solution at the corner points. The algorithms of the solution of problems in rectangular and curvilinear regions, as well as of the Stefan twophase problem, are written. The solution for the rectangular region is represented in an explicit form, whereas the solution for the curvilinear region and the solution of the Stefan problem are reduced to a closed algebraic system of equations of small volume.

Most of the materials undergoing processing in the production processes of chemical technology, from the point of view of the principles of geometry, can be reduced to traditional bodies of canonical shape: a plate, a cylinder, a ball. In the processes of heat treatment of solid materials (heat-moisture treatment, drying, firing), the transfer potentials (temperature, mass content) change significantly over the time of the process. To solve boundary value problems of heat and mass (moisture) conductivity in such cases, the "zonal" method and the "microprocess" method were previously proposed. The possibilities of the "microprocesses" method applied to modeling boundary value problems of heat and mass transfer for bodies of canonical shape under boundary conditions of the first kind (Dirichlet conditions) were shown in the previous works of the authors. This paper presents an illustration of the application of the "microprocess" method for solving boundary value problems of heat and moisture conductivity under more general boundary conditions, conditions of the third kind (Rie-mann-Newton). The great universality of these conditions lies in the fact that, depending on the values of the Biot number (Bi), they transform into a condition of the first kind (Bi tends to zero) or the second (Bi tends to infinity). It is shown that the search for solutions in the region of small values of Fourier numbers (Fo

The paper studies the solvability conditions for one class of boundary value problems for a nonlocal polyharmonic equation in the unit ball with Dirichlet conditions on the boundary generated by a certain orthogonal matrix. The existence and uniqueness of the solution to the posed Dirichlet problem are investigated and the Green's function is constructed. First, some auxiliary statements are established: the inversability of the Vandermonde matrix of the m-th roots of unity is investigated, then the eigenvectors and eigenvalues of the auxiliary matrix generated by the coefficients of the nonlocal operator of the problem are found, and then the inverse matrix to it is obtained. To prove the uniqueness of the solution to the problem, the commutativity of the boundary operators and the nonlocal operator of the problem is established, and it is shown that if a solution to the problem exists, then this solution is a polyharmonic function. Then the conditions for the uniqueness of the solution to the problem under consideration are obtained. Further, on the basis of the auxiliary statements obtained above, conditions for the existence of a solution to the nonlocal problem are found. The solution to this problem is written out through the solution of auxiliary Dirichlet problems for the polyharmonic equation in the unit ball. Finally, using the well-known Green's function of the Dirichlet problem for the polyharmonic equation in the unit ball, the Green's function of the original nonlocal problem is constructed.

A spectral problem occurring in description of small transverse vibrations of a star graph of Stieltjes strings is considered. At all but one pendant vertices Dirichlet conditions are imposed which mean that these vertices are clamped. One vertex (the root) can move with damping in the direction orthogonal to the equilibrium position of the strings. We describe the spectrum of such spectral problem. The corresponding inverse problem lies in recovering the values of point masses and the lengths of the intervals between the masses using the spectrum and some other parameters. We propose conditions on a sequence of complex numbers and a collection of real numbers to be the spectrum of a problem we consider and the lengths of the edges, correspondingly.

The Fourier series (FS) applies to a periodic non-sinusoidal function satisfying the Dirichlet conditions, whereas the being-processed function in practical applications is usually an aperiodic non-sinusoidal signal. When is aperiodic, its calculated FS is not correct, which is still a challenging problem. To overcome the problem, we derive a direct calculation algorithm, a constant iteration algorithm, and an optimal iteration algorithm. The direct calculation algorithm correctly calculates its Fourier coefficients (FCs) when is periodic and satisfies the Dirichlet conditions. Both the constant iteration algorithm and the optimal iteration algorithm provide an idea of determining the states of . From the idea, we obtain an algorithm for determining the states of based on the optimal iteration algorithm. In the algorithm, the variable iteration step is introduced; thus, we present an algorithm for determining the states of based on the variable iteration step. The presented algorithm accurately determines the states of . On the basis of these algorithms, we build a biproportional construction theory. The theory consists of a first and a second proportional construction theory. The former correctly calculates the FCs of at the present sampling time

In this paper, we show Lyapunov and Hartman-Wintner-type inequalities for a fractional partial differential equations with Dirichlet conditions and we give some applications of these inequalities f...

The spectral properties of the nonself-adjoint problem with multipoint perturbations of the Dirichlet conditions for differential operator of order $2n$ with involution are investigated. The system of eigenfunctions of a multipoint problem is constructed. Sufficient conditions have been established, under which this system is complete and, under some additional assumptions, forms the Riesz basis. The research is structured as follows. In section 2 we investigate the properties of the Sturm-type conditions and nonlocal problem with self-adjoint boundary conditions for the equation $$(-1)^ny^{(2n)}(x)+ a_{0}y^{(2n-1)}(x)+ a_{1}y^{(2n-1)}(1-x)=f(x),\,x\in (0,1).$$ In section 3 we study the spectral properties for nonlocal problem with nonself-adjoint boundary conditions for this equation. In sections 4 we construct a commutative group of transformation operators. Using spectral properties of multipoint problem and conditions for completeness the basis properties of the systems of eigenfunctions are established in section 5. In section 6 some analogous results are obtained for multipoint problems generated by differential equations with an involution and are proved the main theorems.

A linear wave equation with acoustic boundary conditions (ABC) on a portion of the boundary and Dirichlet conditions on the rest of the boundary is considered. The (ABC) contain a memory damping with respect to the normal displacement of the boundary point. In this paper, we establish polynomial energy decay rates for the wave equation by using resolvent estimates.

We consider a nonlinear fractional boundary value problem involving conformable variable-order derivative with Dirichlet conditions. We prove the existence of solutions to the considered problem by using the upper and lower solutions’ method with Schauder’s fixed-point theorem. In addition, under some assumptions on the nonlinear term, a new Lyapunov-type inequality is given for the corresponding boundary value problem. The obtained inequality provides a necessary condition for the existence of nontrivial solutions to the considered problem and a method to prove uniqueness for the nonhomogeneous boundary value problem. These new results are illustrated through examples.

This paper focuses on the analysis of a free energy functional, that models a dilute suspension of magnetic nanoparticles in a two-dimensional nematic well. The {\it first part} of the article is devoted to the asymptotic analysis of global energy minimizers in the limit of vanishing elastic constant, $\ell \rightarrow 0$ where the re-scaled elastic constant $\ell$ is inversely proportional to the domain area. The first results concern the strong $H^1$-convergence and a $\ell$-independent $H^2$-bound for the global minimizers on smooth bounded 2D domains, with smooth boundary and topologically trivial Dirichlet conditions. The {\it second part} focuses on the discrete approximation of regular solutions of the corresponding non-linear system of partial differential equations with cubic non-linearity and non-homogeneous Dirichlet boundary conditions. We establish (i) the existence and local uniqueness of the discrete solutions using fixed point argument, (ii) a best approximation result in energy norm, (iii) error estimates in the energy and $L^2$ norms with $\ell $- discretization parameter dependency for the conforming finite element method. Finally, the theoretical results are complemented by numerical experiments on the discrete solution profiles, the numerical convergence rates that corroborates the theoretical estimates, followed by plots that illustrate the dependence of the discretization parameter on $\ell$., Comment: 36 pages, 37 figures

In this paper, we consider the Sturm–Liouville problem with Dirichlet conditions in the case of time scales consists isolated points. Then, we obtain discrete Sturm–Liouville problem on a finite interval. We solve the inverse nodal problem, especially give a reconstruction formula for the potential function q.

In this study, a distributed optimal control problem for n × n cooperative hyperbolic systems with infinite order operators and Dirichlet conditions are considered. The existence and uniqueness of the state of these systems are proved. The necessary and sufficient conditions for optimality of distributed control with constraints are found, and the set of equations and inequalities that defining the optimal control of these systems is also obtained. Finally, some examples for the control problem without constraints are given.

The aim of this paper is to prove the existence of one or multiple solutions of nonlinear difference equations coupled to a general set of boundary conditions. Before to do this, we construct a discrete operator whose fixed points coincide with the solutions of the problem we are looking for. Moreover, we introduce a strong positiveness condition on the related Green’s function that allows us to construct suitable cones where to apply adequate fixed point theorems. Once we have the general existence result, we deduce, as a particular case, the existence of solutions of a second order difference equation with nonlocal perturbed Dirichlet conditions.

A number of problems in the theory of elasticity, the theory of heterogeneous media, the theory of thin shells and plates is reduced to solving boundary value problems for systems of inhomogeneous polyharmonic equations. The paper proposes a numerical algorithm for solving systems of polyharmonic equations of the form in single-connected and multi-connected areas with a piecewise smooth contour with specified boundary conditions. Two cases are considered when the function is a known polyharmonic function and when the function is also the desired polyharmonic function. The boundary conditions can have the form similar to Dirichlet conditions, Neiman conditions, and can have a mixed form when on one part of the boundary conditions of the Dirichlet type are given, and on the other - hand, the conditions of the Neiman type. On the basis of multiple applications of the Laplace operator and the boundary element method, which is based on the green integral identity, the given system is reduced to a system of integral identities. After approximating the boundary by an inscribed n-gon and discretizing the system of integral identities, the latter is reduced to a system of linear algebraic equations, which is conveniently represented as a system of matrix equations. The existence and uniqueness of the solution follows from the existence of a unique solution of a system of linear algebraic equations. Special attention is paid to the application of the algorithm to the solution of problems on the bending of thin plates, and the bending load can be a known function, and can be an unknown polyharmonic function of an arbitrary order with given boundary conditions. The problem of bending a thin plate of elliptic shape with a known load on the surface is solved, as well as the problem of bending a thin square plate with an unknown load, which is the solution of a harmonic equation with given boundary conditions. The level lines are constructed and the forms of curved plates are given.

Under consideration is the identification problem for a time-dependent source in the wave equation. The Dirichlet conditions are used as the boundary conditions, whereas the weighted integral of the conormal derivative of the solution over the boundary of the spatial domain serves as the overdetermination condition. Using the Duhamel method, the problem is reduced to the Volterra integral equation of the first and then the second kind. These results are applied to studying nonlinear coefficient problems. The existence and uniqueness of a local solution is proved by the contraction mapping principle.

We consider the transformation of the initial data space for the Schrodinger equation. The transformation is generated by nonlinear Schrodinger operator on the segment [−π, π] satisfying the homogeneous Dirichlet conditions on the boundary of the segment. The potential here has the type $\xi(u)=\left(1+|u|^{2}\right)^{\frac{p}{2}} u$ , where u is an unknown function, p ≥ 0. The Schrodinger operator defined on the Sobolev space $$H_0^2([-\pi, \pi])$$ generates a vector field $${\rm{v}}:H_0^2([-\pi, \pi])\rightarrow{H}\equiv{L_2}(-\pi, \pi)$$ . First, we study the phenomenon of global existence of a solution of the Cauchy problem for p ∈ [0, 4) and, second, the phenomenon of rise of a solution gradient blow up during a finite time for p ∈ [4, +∞). In second case we study qualitative properties of a solution when it approaches to the boundary of its interval of existence. Moreover, we define a solution extension through the moment of a gradient blow up using both the one-parameter family of multifunctions and the one-parameter family of probability measures on the initial data space of the Cauchy problem. We show that this extension describes the destruction of a solution as the destruction of a pure quantum state and the transition from the set of pure quantum states into the set of mixed quantum states.

Let $$\Omega \subset {\mathbb {R}}^N$$ be an arbitrary open set, $$0

The behavior of solutions of a second-order elliptic equation near a distinguished piece of the boundary is studied. On the remaining part of the boundary, the solutions are assumed to satisfy the homogeneous Dirichlet conditions. A necessary and sufficient condition is established for the existence of an L2 boundary value on the distinguished part of the boundary. Under the conditions of this criterion, estimates for the nontangential maximal function of the solution hold, the solution belongs to the space of (n − 1)-dimensionally continuous functions, and the boundary value is taken in a much stronger sense.