Mixed Type Boundary Value Problems Research Articles

Analytical solution for diffraction of magnetoelastic plane waves by a rigid strip in self-reinforced medium: a contour integration method

This study analytically addresses the diffraction of magnetoelastic plane waves by a rigid strip in a self-reinforced infinite elastic medium. Formulation of the studied model includes mixed type boundary value problem and has been solved by the method of integral equation. The expressions of vertical diffracted displacement and normal stress are established in the closed form by employing the contour integration technique. For the sake of validation, the deduced expressions of vertical diffracted displacement and normal stress are reduced to the special cases, such as without magnetoelastic coupling as well as anisotropy and matched with the pre-established results. The diffraction patterns for the vertical displacement component in the considered material medium are calculated numerically, and its dependence on various affecting parameters viz. magnetoelastic coupling parameter, phase velocity, wave number of the propagating magnetoelastic plane wave, and distance has been depicted graphically for self-reinforced and isotropic materials. Moreover, the influence of anisotropy of the infinite medium has been portrayed remarkably by the comparative study.

Response Mitigation of Floating Platform by Porous-Media-Tuned Liquid Dampers

AbstractA porous-media-tuned liquid damper (PMTLD) can serve as an effective and economical dynamic vibration absorber. Placing porous media within a water tank can improve the capacity for energy dissipation and optimize the performance by varying its material properties. Two numerical models are adopted to simulate the sloshing problem in PMTLD and the dynamics of a floating platform in waves. Besides, the effectiveness of response mitigation can be verified numerically. The first potential-based approach employs a mixed-type boundary value problem (BVP) solver and a free-surface particle tracker. This approach not only simulates the inviscid water wave but also includes the nonlinear damping of the PMTLD via a quadratic Forchheimer term. Another equivalent mechanical model is used to reduce the degree-of-freedom of the PMTLD system. The Newmark method is incorporated to solve the rigid-body dynamics. The second viscous approach uses the finite element method (FEM) to spatially discretize the Navier–Stokes (NS) equations and handles the free surface via the volume of fluid (VOF) and the level set (LS) equations. The multiphase simulation is implemented by computational modeling toolkits, Proteus and Chrono, for the fluid and solid phases, respectively. The correlations between potential flow and two-phase NS models are presented. The PMTLD is designed by analogy with the tuned mass damper (TMD). Numerical results show that the PMTLD can effectively reduce the structure's dynamic response in terms of vibration amplitude around resonance. Such damping devices have great potential for offshore platforms and wind turbine design.

On a degenerate mixed-type boundary value problem for the two-dimensional self-similar Euler equations

On a degenerate mixed-type boundary value problem for the two-dimensional self-similar Euler equations

A mathematical study of diffusive logistic equations with mixed type boundary conditions

<p style='text-indent:20px;'>The purpose of this paper is to provide a careful and accessible exposition of static bifurcation theory for a class of mixed type boundary value problems for diffusive logistic equations with indefinite weights, which model population dynamics in environments with spatial heterogeneity. We discuss the changes that occur in the structure of the positive solutions as a parameter varies near the first eigenvalue of the linearized problem, and prove that the most favorable situations will occur if there is a relatively large favorable region (with good resources and without crowding effects) located some distance away from the boundary of the environment. A biological interpretation of main theorem is that an initial population will grow exponentially until limited by lack of available resources if the diffusion rate is below some critical value; this idea is generally credited to the English economist T. R. Malthus. On the other hand, if the diffusion rate is above this critical value, then the model obeys the logistic equation introduced by the Belgian mathematical biologist P. F. Verhulst. The approach in this paper is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in partial differential equations.</p>

A mixed type boundary–value problem related to the electrostatics of cold plasma jet reactors based on dielectric barrier discharge

A mixed type boundary–value problem related to the electrostatics of cold plasma jet reactors based on dielectric barrier discharge

Sonic-supersonic Solutions to a Mixed-type Boundary Value Problem for the Two-Dimensional Full Euler Equations

We are concerned with the sonic-supersonic structures extracted from the transonic flow problems in gas dynamics. A local classical supersonic solution for the two-dimensional steady full Euler equations is established in an angular region bounded by the sonic curve and the characteristic curve. In order to overcome the challenges caused by the coupling of nonlinearity and degeneracy at the corner point, we develop a new iteration pattern to show the convergence of the iterative sequence generated by the Euler equations in terms of a partial hodograph coordinate system. The pattern developed here will be useful for studying the degenerate mixed-type boundary value problems for other related nonlinear hyperbolic systems.

Mixed type boundary value problems for Laplace–Beltrami equation on a surface with the Lipschitz boundary

Abstract The purpose of the present research is to investigate a general mixed type boundary value problem for the Laplace–Beltrami equation on a surface with the Lipschitz boundary 𝒞 {\mathcal{C}} in the non-classical setting when solutions are sought in the Bessel potential spaces ℍ p s ⁢ ( 𝒞 ) {\mathbb{H}^{s}_{p}(\mathcal{C})} , 1 p < s < 1 + 1 p {\frac{1}{p}<s<1+\frac{1}{p}} , 1 < p < ∞ {1<p<\infty} . Fredholm criteria and unique solvability criteria are found. By the localization, the problem is reduced to the investigation of model Dirichlet, Neumann and mixed boundary value problems for the Laplace equation in a planar angular domain Ω α ⊂ ℝ 2 {\Omega_{\alpha}\subset\mathbb{R}^{2}} of magnitude α. The model mixed BVP is investigated in the earlier paper [R. Duduchava and M. Tsaava, Mixed boundary value problems for the Helmholtz equation in a model 2D angular domain, Georgian Math. J. 27 2020, 2, 211–231], and the model Dirichlet and Neumann boundary value problems are studied in the non-classical setting. The problems are investigated by the potential method and reduction to locally equivalent 2 × 2 {2\times 2} systems of Mellin convolution equations with meromorphic kernels on the semi-infinite axes ℝ + {\mathbb{R}^{+}} in the Bessel potential spaces. Such equations were recently studied by R. Duduchava [Mellin convolution operators in Bessel potential spaces with admissible meromorphic kernels, Mem. Differ. Equ. Math. Phys. 60 2013, 135–177] and V. Didenko and R. Duduchava [Mellin convolution operators in Bessel potential spaces, J. Math. Anal. Appl. 443 2016, 2, 707–731].

On a degenerate mixed-type boundary value problem to the 2-D steady Euler equation

This paper is concerned with the structure of sonic-supersonic solutions near sonic curves arising from the transonic channel flow problems in gas dynamics. Given two smooth curves, one of which is a sonic curve and the other is a characteristic curve, we construct a local classical supersonic solution for the two-dimensional steady Euler equations in the angular region. A novel set of dependent and independent variables are introduced to transform the Euler equations into a linear system with explicitly singularity-regularity structures. We use the iteration method to establish the existence and uniqueness of smooth solutions for the new system and then express the solution in terms of the original variables.

A Boundary Value Problem for a Class of Anisotropic Degenerate Parabolic–Hyperbolic Equations

We consider a mixed type boundary value problem for a class of degenerate parabolic-hyperbolic equations. Namely, we consider a Cartesian product domain and split its boundary into two parts. In one of them we impose a Dirichlet boundary condition; in the other, we impose a Neumann condition. We apply a normal trace formula for $L^2$-divergence-measure fields to prove a new strong trace property in the part of the boundary where the Neumann condition is imposed. We prove existence and uniqueness of the entropy solution. This is a revised corrected version of the paper published in Arch.\ Ration.\ Mech.\ Anal.\ {\bf 226} (2017), no. 3, 975--1008.

A new meshless method for solving steady-state nonlinear heat conduction problems in arbitrary plane domain

For solving steady-state nonlinear heat conduction problems (HCPs) in arbitrary plane domain enclosed by a complex boundary shape, the meshless methods are convenient to programming, of which the polynomial expansion method seems the simplest one. However, it is seldom used as a major medium to solve nonlinear partial differential equations owing to its highly ill-conditioned behavior. First, we decompose the steady-state nonlinear heat conduction equation into a linear portion on the left-hand side, and leave other terms on the right-hand side as a non-homogeneous portion. Then, we need to solve a steady-state linear heat conduction problem in each iteration, for which we propose a multiple-scale Pascal triangle method to generate the linear algebraic equations, where the multiple scales are automatically decided by the collocation points. We can find that these scales can largely reduce the condition number of the coefficient matrix in each linear system, such that the iteration process is convergent very quickly, and the numerical solutions obtained are very accurate and stable against very large noise to over 50%, no matter for the Dirichlet and mixed-type boundary value problems. Numerical results confirm the validity of the present multiple-scale polynomial expansion method for solving steady-state nonlinear HCPs in arbitrary plane domains.

Numerical simulation of Forchheimer flow to a partially penetrating well with a mixed-type boundary condition

This article presents a numerical study to investigate the combined role of partial well penetration (PWP) and non-Darcy effects concerning the performance of groundwater production wells. A finite difference model is developed in MATLAB to solve the two-dimensional mixed-type boundary value problem associated with flow to a partially penetrating well within a cylindrical confined aquifer. Non-Darcy effects are incorporated using the Forchheimer equation. The model is verified by comparison to results from existing semi-analytical solutions concerning the same problem but assuming Darcy’s law. A sensitivity analysis is presented to explore the problem of concern. For constant pressure production, Non-Darcy effects lead to a reduction in production rate, as compared to an equivalent problem solved using Darcy’s law. For fully penetrating wells, this reduction in production rate becomes less significant with time. However, for partially penetrating wells, the reduction in production rate persists for much larger times. For constant production rate scenarios, the combined effect of PWP and non-Darcy flow takes the form of a constant additional drawdown term. An approximate solution for this loss term is obtained by performing linear regression on the modeling results.

Over-determined boundary value problems for PDE and their application in the wave propagation theory

The solvability conditions of the over-determined boundary value problems for PDE with constant coefficients in the form of connection between Fourier transforms of boundary functions are obtained. The jump problem on the hyperplane for PDE is formulated and investigated. Two-dimensional Helmholtz equation is considered as an example. It is proved that the mixed type conjugation problem for Helmholtz equation, to which the diffraction problem of the electromagnetic wave on thin conducting ribbons is reduced, is equivalent to one-sided mixed type boundary value problem. The uniqueness theorem is proved for this problem.

Approximate Solution for a Transient Hydraulic Head Distribution Induced by a Constant-Head Test at a Partially Penetrating Well in a Two-Zone Confined Aquifer

A mathematical model describing the transient hydraulic head distribution induced by constant-head pumping/injection at a partially penetrating well in a radial two-zone confined aquifer is a mixed-type boundary value problem. The analytical solution of the model is in terms of an improper integral with an integrand having a singularity at the origin. The solution should rely on numerical methods to evaluate the integral and handle the problems of convergence and singularity. This study aims at developing a new approximate solution describing the transient hydraulic head distribution for a constant-head test (CHT) at a partially penetrating well in the aquifer. This approximate solution is acquired based on a time-dependent diffusion layer approximation proposed in the field of electrochemistry. The diffusion layer can be analogous to the radius of influence in the area of well hydraulics. The approximate solution is in terms of modified Bessel functions for aquifers with a partially penetrating well and can reduce to a simpler form in terms of a natural logarithmic function for the case of well full penetration. The predicted hydraulic heads from the present approximate solution are compared with those estimated by the Laplace-domain solution of the model. The result shows that the predicted spatial head distributions are accurate in the formation zone and fairly good in the skin zone. In addition, the present solution gives an accurate temporal head distribution at a specific location when the radius of influence is far away from the observation wells. This newly developed approximate solution has advantages of easy computing and good accuracy from practical viewpoint, and thus is a handy tool to evaluate temporal and spatial hydraulic head distributions for the CHT.

Effect of injection screen slot geometry on hydraulic conductivity tests

Summary Hydraulic conductivity and its spatial variability are important hydrogeological parameters and are typically determined through injection tests at different scales. For injection test interpretation, shape factors are required to account for injection screen geometry. Shape factors act as proportionality constants between hydraulic conductivity and observed ratios of injection flow rate and injection head at steady-state. Existing results for such shape factors assume either an ideal screen (i.e., ignoring effects of screen slot geometry) or infinite screen length (i.e., ignoring effects of screen extremes). In the present work, we investigate the combined effects of circumferential screen slot geometry and finite screen length on injection shape factors. This is done in terms of a screen entrance resistance by solving a steady-state potential flow mixed type boundary value problem in a homogeneous axi-symmetric flow domain using a semi-analytical solution approach. Results are compared to existing analytical solutions for circumferential and longitudinal slots on infinite screens, which are found to be identical. Based on an existing approximation, an expression is developed for a dimensionless screen entrance resistance of infinite screens, which is a function of the relative slot area only. For anisotropic conditions, e.g., when conductivity is smaller in the vertical direction than in the horizontal, screen entrance losses for circumferential slots increase, while they remain unaffected for longitudinal slots. This work is not concerned with investigating the effects of (possibly turbulent) head losses inside the injection device including the passage through the injection slots prior to entering the porous aquifer.

A uniformly convergent hybrid scheme for singularly perturbed system of reaction-diffusion Robin type boundary-value problems

This paper deals with the study on system of reaction diffusion differential equations for Robin or mixed type boundary value problems (MBVPs). A cubic spline approximation has been used to obtain the difference scheme for the system of MBVPs, on a piecewise uniform Shishkin mesh defined in the whole domain. It has been shown that our proposed scheme, i.e., central difference approximation for outer region with cubic spline approximation for inner region of boundary layers, leads to almost second order parameter uniform convergence whereas the standard method i.e., the forward-backward approximation for mixed boundary conditions with central difference approximation inside the domain leads to almost first order convergence on Shishkin mesh. Numerical results are provided to show the efficiency and accuracy of these methods.

Coercivity of elliptic mixed boundary value problems in annulus of $\mathbb{R}^N$

In this paper is proved that the Strong Maximum Principle issatisfied for a wide class of linear elliptic boundary valueproblems of mixed type in an annulus of $\mathbb{R}^N$, $N\geq 1$,provided it is thin enough. The coercive character of theseboundary value problems is obtained thanks to the characterizationof the Strong Maximum Principle found in [3], proving thatthe principal eigenvalue associated to each boundary value problemmay be as large as we wish, independently of the weight on theboundary, by taking the annulus thin enough.

A Note on Nonlocal Boundary Value Problems for Hyperbolic Schrödinger Equations

The nonlocal boundary value problem <svg style="vertical-align:-2.34499pt;width:231.66251px;" id="M1" height="16.775" version="1.1" viewBox="0 0 231.66251 16.775" width="231.66251" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.775)"> <g transform="translate(72,-58.58)"> <text transform="matrix(1,0,0,-1,-71.95,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝑑</tspan> </text> <text transform="matrix(1,0,0,-1,-64.55,66.03)"> <tspan style="font-size: 8.75px; " x="0" y="0">2</tspan> </text> <text transform="matrix(1,0,0,-1,-59.67,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝑢</tspan> <tspan style="font-size: 12.50px; " x="5.9264221" y="0">(</tspan> <tspan style="font-size: 12.50px; " x="10.089921" y="0">𝑡</tspan> <tspan style="font-size: 12.50px; " x="14.00336" y="0">)</tspan> <tspan style="font-size: 12.50px; " x="18.166859" y="0">/</tspan> <tspan style="font-size: 12.50px; " x="21.642693" y="0">𝑑</tspan> <tspan style="font-size: 12.50px; " x="28.294289" y="0">𝑡</tspan> </text> <text transform="matrix(1,0,0,-1,-27.47,65.97)"> <tspan style="font-size: 8.75px; " x="0" y="0">2</tspan> </text> <text transform="matrix(1,0,0,-1,-19.81,60.96)"> <tspan style="font-size: 12.50px; " x="0" y="0">+</tspan> <tspan style="font-size: 12.50px; " x="11.327718" y="0">𝐴</tspan> <tspan style="font-size: 12.50px; " x="20.29237" y="0">𝑢</tspan> <tspan style="font-size: 12.50px; " x="26.21879" y="0">(</tspan> <tspan style="font-size: 12.50px; " x="30.38229" y="0">𝑡</tspan> <tspan style="font-size: 12.50px; " x="34.295731" y="0">)</tspan> <tspan style="font-size: 12.50px; " x="41.935062" y="0">=</tspan> <tspan style="font-size: 12.50px; " x="53.975452" y="0">𝑓</tspan> <tspan style="font-size: 12.50px; " x="60.914616" y="0">(</tspan> <tspan style="font-size: 12.50px; " x="65.078117" y="0">𝑡</tspan> <tspan style="font-size: 12.50px; " x="68.991554" y="0">)</tspan> <tspan style="font-size: 12.50px; " x="77.31855" y="0">(</tspan> <tspan style="font-size: 12.50px; " x="81.482048" y="0">0</tspan> <tspan style="font-size: 12.50px; " x="91.209389" y="0">≤</tspan> <tspan style="font-size: 12.50px; " x="103.24977" y="0">𝑡</tspan> <tspan style="font-size: 12.50px; " x="110.63905" y="0">≤</tspan> <tspan style="font-size: 12.50px; " x="122.66693" y="0">1</tspan> <tspan style="font-size: 12.50px; " x="128.91843" y="0">)</tspan> </text> </g> </g> </svg>, <svg style="vertical-align:-2.34499pt;width:243.53751px;" id="M2" height="13.6125" version="1.1" viewBox="0 0 243.53751 13.6125" width="243.53751" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.6125)"> <g transform="translate(72,-61.11)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝑖</tspan> <tspan style="font-size: 12.50px; " x="3.888433" y="0">(</tspan> <tspan style="font-size: 12.50px; " x="8.0519323" y="0">𝑑</tspan> <tspan style="font-size: 12.50px; " x="14.703528" y="0">𝑢</tspan> <tspan style="font-size: 12.50px; " x="20.62995" y="0">(</tspan> <tspan style="font-size: 12.50px; " x="24.793449" y="0">𝑡</tspan> <tspan style="font-size: 12.50px; " x="28.706888" y="0">)</tspan> <tspan style="font-size: 12.50px; " x="32.870388" y="0">/</tspan> <tspan style="font-size: 12.50px; " x="36.346222" y="0">𝑑</tspan> <tspan style="font-size: 12.50px; " x="42.997818" y="0">𝑡</tspan> <tspan style="font-size: 12.50px; " x="46.911255" y="0">)</tspan> <tspan style="font-size: 12.50px; " x="53.850422" y="0">+</tspan> <tspan style="font-size: 12.50px; " x="65.190643" y="0">𝐴</tspan> <tspan style="font-size: 12.50px; " x="74.155296" y="0">𝑢</tspan> <tspan style="font-size: 12.50px; " x="80.081718" y="0">(</tspan> <tspan style="font-size: 12.50px; " x="84.245216" y="0">𝑡</tspan> <tspan style="font-size: 12.50px; " x="88.158653" y="0">)</tspan> <tspan style="font-size: 12.50px; " x="95.797989" y="0">=</tspan> <tspan style="font-size: 12.50px; " x="107.82587" y="0">𝑔</tspan> <tspan style="font-size: 12.50px; " x="113.98985" y="0">(</tspan> <tspan style="font-size: 12.50px; " x="118.15335" y="0">𝑡</tspan> <tspan style="font-size: 12.50px; " x="122.06679" y="0">)</tspan> <tspan style="font-size: 12.50px; " x="130.39378" y="0">(</tspan> <tspan style="font-size: 12.50px; " x="134.55728" y="0">−</tspan> <tspan style="font-size: 12.50px; " x="143.12184" y="0">1</tspan> <tspan style="font-size: 12.50px; " x="152.84918" y="0">≤</tspan> <tspan style="font-size: 12.50px; " x="164.87706" y="0">𝑡</tspan> <tspan style="font-size: 12.50px; " x="172.26633" y="0">≤</tspan> <tspan style="font-size: 12.50px; " x="184.30672" y="0">0</tspan> <tspan style="font-size: 12.50px; " x="190.55823" y="0">)</tspan> </text> </g> </g> </svg>, <svg style="vertical-align:-2.21957pt;width:87.012497px;" id="M3" height="15.3375" version="1.1" viewBox="0 0 87.012497 15.3375" width="87.012497" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,15.3375)"> <g transform="translate(72,-59.73)"> <text transform="matrix(1,0,0,-1,-71.95,61.99)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝑢</tspan> <tspan style="font-size: 12.50px; " x="5.9264221" y="0">(</tspan> <tspan style="font-size: 12.50px; " x="10.089921" y="0">0</tspan> </text> <text transform="matrix(1,0,0,-1,-55.61,67.16)"> <tspan style="font-size: 8.75px; " x="0" y="0">+</tspan> </text> <text transform="matrix(1,0,0,-1,-49.12,61.99)"> <tspan style="font-size: 12.50px; " x="0" y="0">)</tspan> <tspan style="font-size: 12.50px; " x="7.6393328" y="0">=</tspan> <tspan style="font-size: 12.50px; " x="19.667219" y="0">𝑢</tspan> <tspan style="font-size: 12.50px; " x="25.593641" y="0">(</tspan> <tspan style="font-size: 12.50px; " x="29.757139" y="0">0</tspan> </text> <text transform="matrix(1,0,0,-1,-13.1,67.16)"> <tspan style="font-size: 8.75px; " x="0" y="0">−</tspan> </text> <text transform="matrix(1,0,0,-1,-6.61,61.99)"> <tspan style="font-size: 12.50px; " x="0" y="0">)</tspan> </text> </g> </g> </svg>, <svg style="vertical-align:-3.21404pt;width:95.099998px;" id="M4" height="16.575001" version="1.1" viewBox="0 0 95.099998 16.575001" width="95.099998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.575)"> <g transform="translate(72,-58.74)"> <text transform="matrix(1,0,0,-1,-71.95,61.99)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝑢</tspan> </text> <text transform="matrix(1,0,0,-1,-66.02,58.87)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝑡</tspan> </text> <text transform="matrix(1,0,0,-1,-62.79,61.99)"> <tspan style="font-size: 12.50px; " x="0" y="0">(</tspan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0">0</tspan> </text> <text transform="matrix(1,0,0,-1,-52.37,67.16)"> <tspan style="font-size: 8.75px; " x="0" y="0">+</tspan> </text> <text transform="matrix(1,0,0,-1,-45.88,61.99)"> <tspan style="font-size: 12.50px; " x="0" y="0">)</tspan> <tspan style="font-size: 12.50px; " x="7.6268301" y="0">=</tspan> <tspan style="font-size: 12.50px; " x="19.667219" y="0">𝑢</tspan> </text> <text transform="matrix(1,0,0,-1,-20.28,58.87)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝑡</tspan> </text> <text transform="matrix(1,0,0,-1,-17.04,61.99)"> <tspan style="font-size: 12.50px; " x="0" y="0">(</tspan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0">0</tspan> </text> <text transform="matrix(1,0,0,-1,-6.62,67.16)"> <tspan style="font-size: 8.75px; " x="0" y="0">−</tspan> </text> <text transform="matrix(1,0,0,-1,-0.13,61.99)"> <tspan style="font-size: 12.50px; " x="0" y="0">)</tspan> </text> </g> </g> </svg>, <svg style="vertical-align:-2.29482pt;width:131.2625px;" id="M5" height="13.55" version="1.1" viewBox="0 0 131.2625 13.55" width="131.2625" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.55)"> <g transform="translate(72,-61.16)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝐴</tspan> <tspan style="font-size: 12.50px; " x="8.9646511" y="0">𝑢</tspan> <tspan style="font-size: 12.50px; " x="14.891073" y="0">(</tspan> <tspan style="font-size: 12.50px; " x="19.054571" y="0">−</tspan> <tspan style="font-size: 12.50px; " x="27.619127" y="0">1</tspan> <tspan style="font-size: 12.50px; " x="33.870628" y="0">)</tspan> <tspan style="font-size: 12.50px; " x="41.497456" y="0">=</tspan> <tspan style="font-size: 12.50px; " x="53.537846" y="0">𝛼</tspan> <tspan style="font-size: 12.50px; " x="60.589539" y="0">𝑢</tspan> <tspan style="font-size: 12.50px; " x="66.515961" y="0">(</tspan> <tspan style="font-size: 12.50px; " x="70.679459" y="0">𝜇</tspan> <tspan style="font-size: 12.50px; " x="78.293785" y="0">)</tspan> <tspan style="font-size: 12.50px; " x="85.232948" y="0">+</tspan> <tspan style="font-size: 12.50px; " x="96.573174" y="0">𝜑</tspan> </text> </g> </g> </svg>, <svg style="vertical-align:-2.29482pt;width:64.050003px;" id="M6" height="13.55" version="1.1" viewBox="0 0 64.050003 13.55" width="64.050003" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,13.55)"> <g transform="translate(72,-61.16)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0">0</tspan> <tspan style="font-size: 12.50px; " x="9.7148314" y="0">&lt;</tspan> <tspan style="font-size: 12.50px; " x="21.75522" y="0">𝜇</tspan> <tspan style="font-size: 12.50px; " x="32.845383" y="0">≤</tspan> <tspan style="font-size: 12.50px; " x="44.885769" y="0">1</tspan> </text> </g> </g> </svg>, for hyperbolic Schr&#xf6;dinger equations in a Hilbert space <svg style="vertical-align:-0.0pt;width:15.175px;" id="M7" height="10.325" version="1.1" viewBox="0 0 15.175 10.325" width="15.175" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.325)"> <g transform="translate(72,-63.74)"> <text transform="matrix(1,0,0,-1,-71.95,63.79)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝐻</tspan> </text> </g> </g> </svg> with the self-adjoint positive definite operator <svg style="vertical-align:-0.0pt;width:11.325px;" id="M8" height="10.55" version="1.1" viewBox="0 0 11.325 10.55" width="11.325" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.55)"> <g transform="translate(72,-63.56)"> <text transform="matrix(1,0,0,-1,-71.95,63.61)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝐴</tspan> </text> </g> </g> </svg> is considered. The stability estimates for the solution of this problem are established. In applications, the stability estimates for solutions of the mixed-type boundary value problems for hyperbolic Schr&#xf6;dinger equations are obtained.

Existence of solutions for weighted p(r)-Laplacian impulsive system mixed type boundary value problems

Abstract This paper investigates the existence of solutions for weighted p(r)-Laplacian impulsive system mixed type boundary value problems. The proof of our main result is based upon Gaines and Mawhin's coincidence degree theory. Moreover, we obtain the existence of nonnegative solutions.

A new solution and data analysis for gas flow to a barometric pumping well

When an open well is installed in an unsaturated zone, gas will flow to/from the well depending on the pressure difference between the well and the surrounding media. This process is called barometric pumping and the well is called a barometric pumping well (BPW). Understanding subsurface gas pressure distribution and gas flow rate to/from a BPW is indispensable to optimize passive soil vapor extraction. This study presents a 2-D semi-analytical solution to calculate the subsurface gas pressure and gas flow rate to/from a BPW with and without a check valve. The problem is conceptualized as a mixed-type boundary value problem. The solution for pumping without a check valve is used to analyze the behavior of the radius of influence (ROI). Results show that ROI is time-dependent. It increases with radial gas permeability and decreases with vertical gas permeability. Field application of the solution without a check valve demonstrates the high accuracy of the developed solution.

On stable implicit difference scheme for hyperbolic–parabolic equations in a Hilbert space

AbstractThe first‐order of accuracy difference scheme for approximately solving the multipoint nonlocal boundary value problem for the differential equation in a Hilbert space H, with self‐adjoint positive definite operator A is presented. The stability estimates for the solution of this difference scheme are established. In applications, the stability estimates for the solution of difference schemes of the mixed type boundary value problems for hyperbolic–parabolic equations are obtained. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009