CARLEMAN'S FORMULA OF A SOLUTIONS OF THE POISSON EQUATION IN BOUNDED DOMAIN

We suggest an explicit continuation formula for a solution to the Cauchy problem for the Poisson equation in a domain from its values and values of its normal derivative on a part of the boundary. We construct the continuation formula of this problem based on the Carleman--Yarmuhamedov function method.

Bifurcation of a Mathematical Model for Tumor Growth with Angiogenesis and Gibbs–Thomson Relation

In this paper, a mathematical model for solid vascular tumor growth with Gibbs–Thomson relation is studied. On the free boundary, we consider Gibbs–Thomson relation which means energy is expended to maintain the tumor structure. Supposing that the nutrient is the source of the energy, the nutrient denoted by [Formula: see text] satisfies [Formula: see text] where [Formula: see text] is a constant representing the ability of the tumor to absorb the nutrient through its blood vessels; [Formula: see text] is concentration of the nutrient outside the tumor; [Formula: see text] is the mean curvature; [Formula: see text] denotes adhesiveness between cells and [Formula: see text] denotes the exterior normal derivative on [Formula: see text] The existence, uniqueness and nonexistence of radially symmetric solutions are discussed. By using the bifurcation method, we discuss the existence of nonradially symmetric solutions. The results show that infinitely many nonradially symmetric solutions bifurcate from the radially symmetric solutions.

Numerical method for solving the piecewise constant source inverse problem of an elliptic equation from a partial boundary observation data

We present results of numerical investigation of the source term recovery in a boundary value problem for an elliptic equation. An additional information about the solution is considered as its normal derivative taken on a part of the boundary. Such source inverse problem is related with inverse gravimetry problem of determining an inhomogeneity from gravitational potential anomalies on the Earth’s surface. We propose an iterative method for numerical recovery of the source term on the base of minimization of the observation residual by a gradient type method. The numerical implementation is based on finite element approximation using the FEniCS scientific computing platform and the dolfin-adjoint package. The capabilities of the developed computational algorithm are illustrated by results of numerical solutions of two dimensional test problems.

Inverse Problem Related to Boundary Shape Identification for a Hyperbolic Differential Equation

In this paper, we are interested in the inverse problem of the determination of the unknown part ∂ Ω , Γ 0 of the boundary of a uniformly Lipschitzian domain Ω included in ℝ N from the measurement of the normal derivative ∂ n v on suitable part Γ 0 of its boundary, where v is the solution of the wave equation ∂ t t v x , t − Δ v x , t + p x v x = 0 in Ω × 0 , T and given Dirichlet boundary data. We use shape optimization tools to retrieve the boundary part Γ of ∂ Ω . From necessary conditions, we estimate a Lagrange multiplier k Ω which appears by derivation with respect to the domain. By maximum principle theory for hyperbolic equations and under geometrical assumptions, we prove a uniqueness result of our inverse problem. The Lipschitz stability is established by increasing of the energy of the system. Some numerical simulations are made to illustrate the optimal shape.

Important Issues on Spectral Properties of a Transmission Eigenvalue Problem

Nowadays, inverse scattering is an important field of interest for many mathematicians who deal with partial differential equations theory, and the research in inverse scattering is in continuous progress. There are many problems related to scattering by an inhomogeneous media. Here, we study the transmission eigenvalue problem corresponding to a new scattering problem, where boundary conditions differ from any other interior problem studied previously. more specifically, instead of prescribing the difference Cauchy data on the boundary which is the classical form of the problem, we consider the case when the difference of the trace of the fields is proportional to the normal derivative of the field. Typical concerns related to TEP (transmission eigenvalue problem) are Fredholm property and solvability, the discreteness of the transmission eigenvalues, and their existence. In this article, we provide answers for all these concerns in a given interior transmission problem for an inhomogeneous media. We use the variational method and a very important theorem on the existence of transmission eigenvalues to arrive at the conclusion of the existence of the transmission eigenvalues.

Bounded $$H^\infty $$-calculus for a degenerate elliptic boundary value problem

On a manifold X with boundary and bounded geometry we consider a strongly elliptic second order operator A together with a degenerate boundary operator T of the form $$T=\varphi _0\gamma _0 + \varphi _1\gamma _1$$ T = φ 0 γ 0 + φ 1 γ 1 . Here $$\gamma _0$$ γ 0 and $$\gamma _1$$ γ 1 denote the evaluation of a function and its exterior normal derivative, respectively, at the boundary. We assume that $$\varphi _0, \varphi _1\ge 0$$ φ 0 , φ 1 ≥ 0 , and $$\varphi _0+\varphi _1\ge c$$ φ 0 + φ 1 ≥ c , for some $$c>0$$ c > 0 , where either $$\varphi _0,\varphi _1\in C^{\infty }_b(\partial X)$$ φ 0 , φ 1 ∈ C b ∞ ( ∂ X ) or $$\varphi _0=1 $$ φ 0 = 1 and $$\varphi _1=\varphi ^2$$ φ 1 = φ 2 for some $$\varphi \in C^{2+\tau }(\partial X)$$ φ ∈ C 2 + τ ( ∂ X ) , $$\tau >0$$ τ > 0 . We also assume that the highest order coefficients of A belong to $$C^\tau (X)$$ C τ ( X ) and the lower order coefficients are in $$L_\infty (X)$$ L ∞ ( X ) . We show that the $$L_p(X)$$ L p ( X ) -realization of A with respect to the boundary operator T has a bounded $$H^\infty $$ H ∞ -calculus. We then obtain the unique solvability of the associated boundary value problem in adapted spaces. As an application, we show the short time existence of solutions to the porous medium equation.

A High-Order Flux Reconstruction Method for 2-D Vorticity Transport

A compact high-order finite difference method on unstructured meshes is developed for discretization of the unsteady vorticity transport equations (VTE) for 2-D incompressible flow. The algorithm is based on the Flux Reconstruction Method of Huynh [1, 2], extended to evaluate a Poisson equation for the streamfunction to enforce the kinematic relationship between the velocity and vorticity fields while satisfying the continuity equation. Unlike other finite difference methods for the VTE, where the wall vorticity is approximated by finite differencing the second wall-normal derivative of the streamfunction, the new method applies a Neumann boundary condition for the diffusion of vorticity such that it cancels the slip velocity resulting from the solution of the Poisson equation for the streamfunction. This yields a wall vorticity with order of accuracy consistent with that of the overall solution. In this paper, the high-order VTE solver is formulated and results presented to demonstrate the accuracy and convergence rate of the Poisson solution, as well as the VTE solver using benchmark problems of 2-D flow in lid-driven cavity and backward facing step channel at various Reynolds numbers.

Приближенное представление дилогарифмами решения одной вариационной краевой задачи для круга при граничном условии Неймана

It is known that boundary value problems for the Laplace and Poisson equations are equivalent to the problem of the calculus of variations – the minimum of an integral for which the given partial differential equation is the Euler – Lagrange equation. For example, the problem of the minimum of the Dirichlet integral in the unit disc centered at the origin on some admissible set of functions for given values of the normal derivative on the circle is equivalent to the Neimann boundary value problem for the Laplace equation in this domain. An effective approximate dilogarithm representation of the solution of the above equivalent variational boundary value problem is constructed on the basis of the known exact solution of the Neumann Boundary value problem for a circle using a special approximate formula for the Dini integral. The approximate formula is effective in the sense that it is quite simple in numerical implementation, stable, and the error estimation, which is uniform over a circle, allows calculations with the given accuracy. A special quadrature formula for the Dini integral has a remarkable property – its coefficients are non-negative. Quadrature formulas with non-negative coefficients occupy a special place in the theory of approximate calculations of definite integrals and its applications. Naturally, this property becomes even more significant when the coefficient are not number, but some functions. The performed numerical analysis of the approximate solution confirms its effectiveness.

Analysis of reaction–diffusion systems where a parameter influences both the reaction terms as well as the boundary

We study positive solutions to steady-state reaction–diffusion models of the form $$ \textstyle\begin{cases} -\Delta u=\lambda f(v);\quad\Omega, \\ -\Delta v=\lambda g(u);\quad\Omega, \\ \frac{\partial u}{\partial \eta }+\sqrt{\lambda } u=0;\quad \partial \Omega, \\ \frac{\partial v}{\partial \eta }+\sqrt{\lambda }v=0; \quad\partial \Omega, \end{cases} $$ { − Δ u = λ f ( v ) ; Ω , − Δ v = λ g ( u ) ; Ω , ∂ u ∂ η + λ u = 0 ; ∂ Ω , ∂ v ∂ η + λ v = 0 ; ∂ Ω , where $\lambda >0$ λ > 0 is a positive parameter, Ω is a bounded domain in $\mathbb{R}^{N}$ R N $(N > 1)$ ( N > 1 ) with smooth boundary ∂Ω, or $\Omega =(0,1)$ Ω = ( 0 , 1 ) , $\frac{\partial z}{\partial \eta }$ ∂ z ∂ η is the outward normal derivative of z. We assume that f and g are continuous increasing functions such that $f(0) = 0 = g(0)$ f ( 0 ) = 0 = g ( 0 ) and $\lim_{s \rightarrow \infty } \frac{f(Mg(s))}{s} = 0$ lim s → ∞ f ( M g ( s ) ) s = 0 for all $M>0$ M > 0 . In particular, we extend the results for the single equation case discussed in (Fonseka et al. in J. Math. Anal. Appl. 476(2):480-494, 2019) to the above system.

A second-order accurate structure-preserving scheme for the Cahn-Hilliard equation with a dynamic boundary condition

<p style='text-indent:20px;'>We propose a structure-preserving finite difference scheme for the Cahn–Hilliard equation with a dynamic boundary condition using the discrete variational derivative method (DVDM) proposed by Furihata and Matsuo [<xref ref-type="bibr" rid="b14">14</xref>]. In this approach, it is important and essential how to discretize the energy which characterizes the equation. By modifying the conventional manner and using an appropriate summation-by-parts formula, we can use a standard central difference operator as an approximation of an outward normal derivative on the discrete boundary condition of the scheme. We show that our proposed scheme is second-order accurate in space, although the previous structure-preserving scheme proposed by Fukao–Yoshikawa–Wada [<xref ref-type="bibr" rid="b13">13</xref>] is first-order accurate in space. Also, we show the stability, the existence, and the uniqueness of the solution for our proposed scheme. Computation examples demonstrate the effectiveness of our proposed scheme. Especially through computation examples, we confirm that numerical solutions can be stably obtained by our proposed scheme.</p>