Showing papers on "Laplace's equation published in 2022"
TL;DR: In this article , the authors developed analytical methods for the calculation of blockage coefficients, which characterize the presence of an object on uniform flow along a rigid pipe, and showed that the blockage coefficient is related to the added mass of the object moving along a pipe.
Abstract: Blockage coefficients characterize the presence of an object on uniform flow along a rigid pipe. The main purpose of the paper is to develop analytical methods for the calculation of these coefficients. For potential flow, blockage coefficients are related to the added mass of an object moving along a pipe. They also arise when matched asymptotic expansions are used to solve acoustic problems, in which Laplace’s equation is appropriate for related inner problems. Steady diffusion problems can also lead to similar harmonic problems. The paper includes some general results (such as a version of Webster’s horn equation for slender rigid objects) and some specific results for objects of various simple axisymmetric shapes (spheroids, spindles, finite circular cylinders, double cones and thin discs). Applications to various physical situations are expected in future work.
5 citations
TL;DR: In this paper , a fractional-order wave equation is established and solved for a space of three dimensions using spherical coordinates, where the acoustic wave propagates only in the fluid saturating the porous medium; this model is a special case of Biot's theory obtained by the symmetry of the Lagrangian.
Abstract: A fractional-order wave equation is established and solved for a space of three dimensions using spherical coordinates. An equivalent fluid model is used in which the acoustic wave propagates only in the fluid saturating the porous medium; this model is a special case of Biot’s theory obtained by the symmetry of the Lagrangian (invariance by translation and rotation). The basic solution of the wave equation is obtained in the time domain by analytically calculating Green’s function of the porous medium and using the properties of the Laplace transforms. Fractional derivatives are used to describe, in the time domain, the fluid–structure interactions, which are of the inertial, viscous, and thermal kind. The solution to the fractional-order wave equation represents the radiation field in the porous medium emitted by a point source. An important result obtained in this study is that the solution of the fractional equation is expressed by recurrence relations that are the consequence of the modified Bessel function of the third kind, which represents a physical solution of the wave equation. This theoretical work with analytical results opens up prospects for the resolution of forward and inverse problems allowing the characterization of a porous medium using spherical waves.
5 citations
TL;DR: The Young-Laplace (Y-L) equation as mentioned in this paper relates the pressure difference across the interface of two fluids (such as air and water) to the curvature of the interface.
Abstract: The Young-Laplace (Y-L) equation relates the pressure difference across the interface of two fluids (such as air and water) to the curvature of the interface. The pressure rises on crossing a convex interface such as a rain drop and falls on crossing a concave interface such as the meniscus of water in a glass capillary. The relation between surface geometry and pressure difference across the interface provides the key concept in understanding a large collection of phenomena in hydrostatics. Of the many phenomena in hydrostatics that can be explained readily by the application of the Y-L equation, we consider the ones that are of particular interest in the introductory physics courses. These include the differential pressure within soap bubbles and liquid droplets, the rise and fall of liquids in capillaries, and the depth of liquid spills.
4 citations
TL;DR: In this article, a parabolic p ( x ) -Laplace equation with logarithmic source u q (x ) log u was considered and the singular properties of solutions were determined completely by classifying the initial energy.
Abstract: This paper deals with a parabolic p ( x ) -Laplace equation with logarithmic source u q ( x ) log u . The singular properties of solutions are determined completely by classifying the initial energy. Moreover, we obtain a new extinction rate of solutions, where the order of the extinction rate is greater than the maximum of variable exponent q ( x ) . This kind of extinction rate could reflect the influence of logarithmic functions on the extinction of solutions more reasonably.
4 citations
TL;DR: In this article , the derivation assumptions of the Young-Laplace equation are evaluated closely to discuss the possible modifications towards making conclusive remarks about the predictive power of the equation at the nano-scale.
Abstract: Deductive arguments regarding the unexpected stability of nanobubbles in water include the excessive internal pressure of minuscule gas pockets. In this study, the derivation assumptions of the Young–Laplace equation are evaluated closely to discuss the possible modifications towards making conclusive remarks about the predictive power of the equation at the nano-scale.
4 citations
TL;DR: In this article , the authors show how wettability measurements under conditions of undersaturated water vapor affect the observed water droplet parameters, such as contact angles, surface tension, droplet surface area and droplet volume.
Abstract: Performing digital video image processing of sessile droplets and Laplace fit optimization is one of the most widely used modern techniques to measure the contact angle. The attractive feature of this approach is related to the possibility of simultaneous determination of numerous droplet parameters, such as contact angles, droplet surface tension, droplet surface area and droplet volume. However, lately in the literature, the analysis of the droplet shape is usually restricted by the contact angle determination, and the experiments are performed under open atmospheric conditions without simultaneous monitoring of the other droplet parameters. In this brief note, it is shown how wettability measurements under conditions of undersaturated water vapor affect the observed water droplet parameters.
4 citations
TL;DR: In this paper , a localized virtual boundary element-meshless collocation method (LVBE-MCM) is proposed to solve Laplace and Helmholtz equations in complex two-dimensional (2D) geometries.
Abstract: A localized virtual boundary element–meshless collocation method (LVBE-MCM) is proposed to solve Laplace and Helmholtz equations in complex two-dimensional (2D) geometries. “Localized” refers to employing the moving least square method to locally approximate the physical quantities of the computational domain after introducing the traditional virtual boundary element method. The LVBE-MCM is a semi-analytical and domain-type meshless collocation method that is based on the fundamental solution of the governing equation, which is different from the traditional virtual boundary element method. When it comes to 2D problems, the LVBE-MCM only needs to calculate the numerical integration on the circular virtual boundary. It avoids the evaluation of singular/strong singular/hypersingular integrals seen in the boundary element method. Compared to the difficulty of selecting the virtual boundary and evaluating singular integrals, the LVBE-MCM is simple and straightforward. Numerical experiments, including irregular and doubly connected domains, demonstrate that the LVBE-MCM is accurate, stable, and convergent for solving both Laplace and Helmholtz equations.
3 citations
TL;DR: In this paper , a mathematical approach based on Laplace equation is used to calculate the potential distribution in a two-dimensional rectangle, and a piezoresistivity theory is derived that takes into account changes in dimensions as well as longitudinal and transverse resistivity.
3 citations
TL;DR: In this article , initial boundary value problems with a time-nonlocal condition for a subdiffusion equation with the Riemann-Liouville time-fractional derivatives are considered.
Abstract: Initial boundary value problems with a time-nonlocal condition for a subdiffusion equation with the Riemann-Liouville time-fractional derivatives are considered. The elliptical part of the equation is the Laplace operator, defined in an arbitrary N−dimensional domain Ω with a sufficiently smooth boundary ∂Ω. The existence and uniqueness of the solution to the considered problems are proved. Inverse problems are studied for determining the right-hand side of the equation and a function in a time-nonlocal condition. The main research tool is the Fourier method, so the obtained results can be extended to subdiffusion equations with a more general elliptic operator.
3 citations
TL;DR: In this paper , an enhanced Lo-Christensen-Wu (LCW) theory is defined in the Laplace domain to predict the thermo-mechanical-viscoelastic behavior of long-term composite structures.
Abstract: Abstract An enhanced Lo–Christensen–Wu (LCW) theory is defined in the Laplace domain to predict the thermo-mechanical-viscoelastic behavior of long-term composite structures. The primary objective herein is to systematically extract the computational benefits of the conventional LCW and fifth-order zigzag model via the mixed variational theorem (MVT). Furthermore, the Laplace transform is employed to circumvent the numerical complexity of viscoelastic analysis. The relationships between the two fields were derived using the MVT constraint equations in the Laplace domain. Consequently, the proposed theory has the C0-based computational benefits as the conventional LCW, while improving the solution accuracy for long-term thermo-mechanical-viscoelastic behaviors.
3 citations
TL;DR: In this paper , the authors used axisymmetric drop shape analysis (ADSA) to measure the deviation of the drop shape from purely Laplacian, following the Young-Laplace equation.
Abstract: Interfacial tension and dilatational rheology are often used to characterize the mechanical response of a liquid interface using axisymmetric drop shape analysis (ADSA). It is important to note that for systems dominated by adsorption/desorption of surfactants, the contributions of extra mechanical stresses are negligible; thus, the Young-Laplace equation remains valid. However, for interfaces dominated by extra stresses, as in the case of particle monolayers or asphaltenes that clearly exhibit a skin (a rigid film), the nature of the elastic response is fundamentally different and the validity of the equation is questionable. Calculation of the interfacial tension and dilatational elasticity using drop shape analysis depends critically on the drop shape following the Young-Laplace equation. If the interface becomes more like a solid, the drop shape will deviate from being purely Laplacian. Indeed, the drop will exhibit a wrinkled surface as collapse continues. The geometric parameter RV/A, defined as the ratio (dV/V)/(dA/A) with V is the volume of the drop and A is the area of the interface), allows one to measure the deviation of the drop shape from purely Laplacian. For a simple interface (pure liquids or surfactant solutions), RV/A is quite close to the theoretical value of 1.5 of a perfect sphere. Nevertheless, if the molecules adsorbed at the interface begin to interact strongly, the ratio can vary. In the limit of long-time-scale experiments, RV/A of some drops approaches 2. We studied the evolution of the parameter RV/A for different systems, from simple to complex, as a function of oscillation frequencies and amplitudes of drop volume. The results obtained were compared to the values of the interfacial moduli and drop shape behavior to better characterize the regime change.
TL;DR: In this paper , a modified and generalized form of the KdV-mKdV equation is presented. But the authors do not consider the effects of the fractional order on the wave dynamics of the proposed equation.
Abstract: The KdV equation has many applications in mechanics and wave dynamics. Therefore, researchers are carrying out work to develop and analyze modified and generalized forms of the standard KdV equation. In this paper, we inspect the KdV-mKdV equation, which is a modified and generalized form of the ordinary KdV equation. We use the fractional operator in the Caputo sense to analyze the equation. We examine some theoretical results concerned with the solution’s existence, uniqueness, and stability. We employ a modified Laplace method to extract the numerical results of the considered equation. We use MATLAB-2020 to simulate the results in a few fractional orders. We report the effects of the fractional order on the wave dynamics of the proposed equation.
TL;DR: Theorem 4.3 as discussed by the authors gives a mixed Dirichlet-Neumann problem for the Laplace equation in the complement of a bounded convex polygonal quadrilateral in the extended complex plane.
Abstract: We study the mixed Dirichlet–Neumann problem for the Laplace equation in the complement of a bounded convex polygonal quadrilateral in the extended complex plane. The Dirichlet / Neumann conditions at opposite pairs of sides are $$\{0,1\}$$ and $$\{0,0\},$$ resp. The solution to this problem is a harmonic function in the unbounded complement of the polygon known as the potential function of the quadrilateral. We compute the values of the potential function u including its value at infinity. The main result of this paper is Theorem 4.3 which yields a formula for $$u(\infty )$$ expressed in terms of the angles of the polygonal given quadrilateral and the well-known special functions. We use two independent numerical methods to illustrate our result. The first method is a Mathematica program and the second one is based on using the MATLAB toolbox PlgCirMap. The case of a quadrilateral, which is the exterior of the unit disc with four fixed points on its boundary, is considered as well.
TL;DR: The interval parametric integral equation system (interval PIES) as mentioned in this paper was developed for solving boundary problems with input data defined in this way, and the reliability of the interval PIES solutions obtained using such arithmetic was verified on 2D problems described by Laplace's equation.
TL;DR: In this paper , the backstepping method with a Fredholm transformation was used to stabilize the heat equation on the 1-dimensional torus using the backstacking method with the Laplacian operator.
TL;DR: In this article , a method to calculate the meniscus shape by solving the differential equation based on the Young-Laplace equation is presented, which is solved by applying the cubic Bézier curve.
Abstract: This work presents a method to calculate the meniscus shape by solving the differential equation based on the Young–Laplace equation. More specifically, the differential equation is solved by applying the cubic Bézier curve. A complicated nonlinear differential equation is solved using the Bézier control points and the least-squares method while maintaining computational simplicity. The results show all of the expected features of the meniscus under the gravitational force. A brief discussion is also made on the effect of the errors on the results. The method is further validated by its agreement with the numerical solutions reported in the existing literature.
TL;DR: In this paper , a meshless rule-based method for the determination of myocardial fiber arrangement without requiring a mesh discretization as it is required by FEM has been proposed.
TL;DR: In this paper , a fast method of high order approximations for the solution of the stationary thermoelastic system in ℝ3 in the unknown displacement vector u and temperature T was proposed.
Abstract: We propose a fast method of high order approximations for the solution of the stationary thermoelastic system in ℝ3 in the unknown displacement vector u and temperature T. The problem of determining T is an independent problem of u and can be obtained by solving a Dirichlet problem for the Laplace equation. When the temperature is known, the displacement u is obtained by solving the Lamé system, where the gradient of T is treated as a mass force. Using the basis functions introduced in the theory of approximate approximations, we derive fast and accurate high order formulas for the approximation of u and T. The high accuracy of the method and the convergence order 2, 4, 6, and 8 are confirmed by numerical experiments.
TL;DR: In this article , the fractional Laplace-Adomian decomposition method is applied to obtain the approximate numerical solutions of the heat equation with a heat source and heat loss, and the graphical representations of the solutions depending on the order of fractional derivatives are given.
Abstract: In this paper, we are interested in obtaining an approximate numerical solution of the fractional heat equation where the fractional derivative is in Caputo sense. We also consider the heat equation with a heat source and heat loss. The fractional Laplace-Adomian decomposition method is applied to gain the approximate numerical solutions of these equations. We give the graphical representations of the solutions depending on the order of fractional derivatives. Maximum absolute error between the exact solutions and approximate solutions depending on the fractional-order are given. For the last thing, we draw a comparison between our results and found ones in the literature.
TL;DR: In this paper , the authors developed a walk-on-sphere method for fractional Poisson equations with Dirichilet boundary conditions in high dimensions, which is based on probabilistic representation of the fractional poisson equation.
Abstract: We develop walk‐on‐sphere method for fractional Poisson equations with Dirichilet boundary conditions in high dimensions. The walk‐on‐sphere method is based on probabilistic representation of the fractional Poisson equation. We propose efficient quadrature rules to evaluate integral representation in the ball and apply rejection sampling method to drawing from the computed probabilities in general domains. Moreover, we provide an estimate of the number of walks in the mean value for the method when the domain is a ball. We show that the number of walks is increasing in the fractional order and the distance of the starting point to the origin. We also give the relationship between the Green function of fractional Laplace equation and that of the classical Laplace equation. Numerical results for problems in 2–10 dimensions verify our theory and the efficiency of the modified walk‐on‐sphere method.
TL;DR: In this paper , the authors proposed a new algorithm to remove the ill-conditioning of the classical MFS in the context of the Laplace equation defined in planar domains.
Abstract: The method of fundamental solutions (MFS) is a numerical method for solving boundary value problems involving linear partial differential equations. It is well-known that it can be very effective assuming regularity of the domain and boundary conditions. The main drawback of the MFS is that the matrices involved are typically ill-conditioned and this may prevent the method from achieving high accuracy. In this work, we propose a new algorithm to remove the ill-conditioning of the classical MFS in the context of the Laplace equation defined in planar domains. The main idea is to expand the MFS basis functions in terms of harmonic polynomials. Then, using the singular value decomposition and Arnoldi orthogonalization, we define well conditioned basis functions spanning the same functional space as the MFS’s. Several numerical examples show that when possible to be applied, this approach is much superior to previous approaches, such as the classical MFS or the MFS-QR.
TL;DR: In this paper , positive solutions to semilinear elliptic problems with Hardy potential and a first order term in bounded smooth domain $ \Omega $ with $ 0\in \overline
Abstract: We consider positive solutions to semilinear elliptic problems with Hardy potential and a first order term in bounded smooth domain $ \Omega $ with $ 0\in \overline \Omega $. We deduce symmetry and monotonicity properties of the solutions via the moving plane procedure under suitable assumptions on the nonlinearity.
TL;DR: In this paper , a field-view model is proposed to theoretically analyze the physical mechanism of TENGs, which reveals high efficiency and great accuracy. And the dynamic behavior of Tengs in a single approaching/separating cycle could be regarded as a time series of electrostatic equilibrium problems.
Abstract: Further investigation of theoretical models is essential to physically understand triboelectric nanogenerators (TENGs) and then promote their extensive applications. Theoretical approaches in the existing studies mainly focus on establishing equivalent circuits, which provide a rapid analysis method but with relatively low precision. Therefore, we propose a field-view model to theoretically analyze the physical mechanism of TENGs, which reveals high efficiency and great accuracy. After the comprehensive study, it is determined that the dynamic behavior of TENGs in a single approaching/separating cycle could be regarded as a time series of electrostatic equilibrium problems. Thus, based on Laplace's equations and potential boundary conditions, a definite-solution-problem model is developed, which could provide explicit mathematical expressions to ultra-precisely predict the electrical characteristics of TENG.
TL;DR: In this article , a universal Z 1 transformation was proposed to obtain the general solution of the Laplace equation for the first time, and the form of the general solutions of some PDEs is not unique.
Abstract: With the purpose of concisely and effectively obtaining the general or exact solutions of partial differential equations (PDEs), we put forward some universal Z 1 transformations in present paper. Not only many linear equations can be solved, but also analytical solutions of some nonlinear equations can be obtained by utilizing this method, and many solutions contain arbitrary functions. Taking as the typical case, we gain the general solution of Laplace equation for the first time. During the solving process, we find that the form of the general solution of some PDEs is not unique. On the basis of the practical cases, we also find that general solutions of some first-order linear PDEs obtained by the characteristic equation method are incomplete.
TL;DR: In this article , a semi-analytical method is proposed to analyze the dynamic behavior of horizontal cylindrical shells partially filled with liquid, considering the sloshing effect of the free surface.
Abstract: In this paper, a semi-analytical method is proposed to analyze the dynamic behavior of horizontal cylindrical shells partially filled with liquid, considering the sloshing effect of the free surface. Two coordinate systems are set at the midpoint of the free liquid surface and the geometric center of the cylindrical shell’s cross-section, respectively. The internal fluid is an inviscid, irrotational, and incompressible fluid. The liquid potential functions which satisfy the Laplace function are described in the anti-symmetrical/symmetrical forms based on the liquid surface coordinate system. Meanwhile, the coupled motion functions of the shell are established on the structural coordinate system using the Flugge shell theory. Through the continuous condition on the internal wet surface, the coupled system’s governing equations are achieved and solved by the coordinate transformation and the Galerkin method. The fluid sloshing frequencies and the coupled vibration frequencies of the shell are simultaneously obtained in this coupled model. The accuracy of this method is verified by the published data and the finite element method. Furthermore, the influences of the coupled system’s parameters on the shell’s natural frequencies and sloshing frequencies are discussed, and the coupling effect is revealed between the shell’s vibration and the sloshing of the free surface.
TL;DR: In this article , a numerical method has been proposed for the solution of the IBVP Laplace equation with initial boundary conditions based on finite difference methods and the stability of the difference schemes are guaranteed.
Abstract: In this study, Laplace partial differential equations with initial boundary conditions has been studied. A numerical method has been proposed for the solution of the IBVP Laplace equation. The technique based on finite difference methods. The stability of the difference schemes are guaranteed. Approximation solution of the problem was achieved. For testing the accuracy of the proposed method, two different initial boundary value problems are provided. Moreover, a comparison between the numerical solution and analytical solution has been done. MATLAB software implemented for calculation of absolute errors. Illustration graphs presented. It has been demonstrated that the results of the comparison guarantee the accuracy and reliability of the provided method.
TL;DR: In this article , a strong maximum principle for weak solutions of the mixed local and non-local p-Laplace equation was established for weak weak solutions, where Ω ⊂ R N is an open set, p ∈ ( 1, ∞ ), s ∈( 0, 1 ) and c ∈ C ( Ω ‾ ).
Abstract: We establish a strong maximum principle for weak solutions of the mixed local and nonlocal p-Laplace equation − Δ p u + ( − Δ ) p s u = c ( x ) | u | p − 2 u in Ω , where Ω ⊂ R N is an open set, p ∈ ( 1 , ∞ ), s ∈ ( 0 , 1 ) and c ∈ C ( Ω ‾ ).