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We study second-order consensus dynamics with random additive disturbances. To quantify the robustness of these networks, we investigate three different performance measures: the steady-state variance of pairwise differences between vertex states, the steady-state variance of the deviation of each vertex state from the average, and the total steady-state variance of the system. We show that these performance measures are closely related to the concept of biharmonic distance; the square of the biharmonic distance plays a similar role in the system performance as resistance distance plays in the performance of first-order noisy consensus dynamics. We then define the new concepts of biharmonic Kirchhoff index and vertex centrality based on the biharmonic distance. We further derive analytical results for the performance measures and concepts for complete graphs, star graphs, cycles, and paths, and we use this analysis to compare the asymptotic behavior of the steady-state variance in first-and second-order systems. Finally, we propose a theoretically guaranteed approximation algorithm to estimate the total steady-state variance, which has a complexity of nearly linear time with respect to the number of edges. Extensive experiments results validate both efficiency and accuracy of our algorithm.

In this paper, a meshless discrete scheme based on the generalized finite difference method (GFDM) is proposed to solve the biharmonic interface problem. This scheme turns the interface problem to be two non-interface subproblems coupled with the interface conditions. The interface conditions act as the boundary conditions of the subproblems because the interface plays the role of their boundary. Therefore, the proposed method has the advantage of dealing with problems with complex geometrical interfaces. Due to the GFDM approximates the derivatives of the unknown variables by using a linear combination of the values for nearby nodes, the proposed method also has the advantage in dealing with the interface condition with the jump of derivatives. Four numerical examples are provided to verify the accuracy and the stability of the GFDM for biharmonic interface problems. The numerical results show that the H1 errors can reach almost the same convergence rate of the L∞ and L2 errors. The convergence rates are almost and even higher than fourth-order when the fourth-order Taylor Series expansion is adopted in the GFDM.

The second moment of inertia of a continuous axisymmetric object with an arbitrary shape made of a nonhomogeneous material is usually calculated by dividing it into small domains. However, it is a burdensome process to specify the density of the small domains. In this paper, a technique of easily calculating the second moment of inertia of an axisymmetric nonhomogeneous material using boundary integral equations is proposed. The calculations of the mass, primary moment, and center of mass of an arbitrarily shaped object made of a nonhomogeneous material are also shown. A formulization of the boundary element method is utilized, and a technique for the direct numerical integration of the axisymmetric domain using an axisymmetric interpolation method without the need to carry out domain division is proposed. Heterogeneous materials include laminated material composites, in which the density distribution is discontinuous. Axisymmetric interpolation using harmonic and biharmonic functions has a weak point that can be easily overcome using a scale method.

In this article, we study the existence of solutions for nonlocal p x -biharmonic Kirchhoff-type problem with Navier boundary conditions. By different variational methods, we determine intervals of parameters for which this problem admits at least one nontrivial solution.

Differential equations with variable exponent arise from the nonlinear elasticity theory and the theory of electrorheological fluids. We study the existence of at least three weak solutions for the nonlocal elliptic problems driven by p ⁢ ( x ) p(x) -biharmonic operator. Our technical approach is based on variational methods. Some applications illustrate the obtained results. We also provide an example in order to illustrate our main abstract results. We extend and improve some recent results.

In this paper, we consider the well-posedness of the inhomogeneous nonlinear biharmonic Schrodinger equation with spatial inhomogeneity coefficient K ( x ) behaves like | x | − b for 0 b min ⁡ { N 2 , 4 } . We show the local well-posedness in the whole H s -subcritical case, with 0 s ≤ 2 . The difficulties of this problem come from the singularity of K ( x ) and the lack of differentiability of the nonlinear term. To resolve this, we derive the bilinear Strichartz's type estimates for the nonlinear biharmonic Schrodinger equations in Besov spaces.

In this paper, we study the inverse problem of identifying unknown sources for nonhomogeneous Biharmonic Equations in bounded domain. It shows that the problem is ill-posed. We use Landweber iterative regularization method to recover the ill-posed solution and give the convergence estimates under the priori and posteriori regularization parameter selection rules. Finally, several numerical examples are given to show the effectiveness and stability of this method.

Modelling and simulation of solar power potential (SPP, Wm−2) is dominated by non-differential models, which are prone to truncation errors. However, this work focuses on presenting a more precise SPP model through a biharmonic differential model (BDM). The exact solution of the BDM ensures an accurate prediction of SPP in the study area (Uganda). This is possible by transforming the Mercatorian coordinates into Cartesian coordinates and establishing closed boundary conditions describing the SPP (Wm−2). Single- and multiple-order-exact solutions of the BDM are presented. The body validation of the simulated results is carried out using National Aeronautic and Space Administration (NASA) quadrennial SPP satellite data (1984–2019). The study is conducted on Regional and Non-regional basis to identify areas with high SPP values. Validation of the simulated SPP results indicates that multiple-order-exact solution is more accurate than the single-order-exact solution based on the mean square error (RMSE) analysis. The SPP (dense area km2, percentage area %) for the Non-regional, Northern Region, Eastern Region, Central Region and Western Region is as follows: 870 (170,817, 7.37), 831.4 (20,250, 23.71), 829.2 (10941.6, 27.71), 823.0 (2677.2, 4.36) and 798.0 (13,600, 0.73), respectively, were investigated. Therefore, it is clear that the BDM could be used to model SPP field data using the multiple-order-exact model solution. Thus, BDM produces a better result than non-differential models by eliminating truncation error in its solution. Therefore, for the purposes of estimating accurate information on the SPP, the BDM and its exact multiple-order solution should be applied.

We construct infinitely many real-valued, time-periodic breather solutions of the nonlinear wave equation ∂ t t U − Δ U = Q ( x ) | U | p − 2 U on T × R N with suitable N ⩾ 2, p > 2 and localized nonnegative Q. These solutions are obtained from critical points of a dual functional and they are weakly localized in space. Our abstract framework allows to find similar existence results for the nonlinear Klein–Gordon equation and biharmonic wave equations.

In this paper we are going to investigate the Lichnerowicz-type estimate theorem for the first eigenvalue of buckling and clamped plate problems on both Riemannian and Kahler manifolds. Also we wil...

Seeking efficient solutions to nonlinear boundary value problems is a crucial challenge in the mathematical modelling of many physical phenomena. A well-known example of this is solving the Biharmonic equation relating to numerous problems in fluid and solid mechanics. One must note that, in general, it is challenging to solve such boundary value problems due to the higher-order partial derivatives in the differential operators. An artificial neural network is thought to be an intelligent system that learns by example. Therefore, a well-posed mathematical problem can be solved using such a system. This paper describes a mesh free method based on a suitably crafted deep neural network architecture to solve a class of well-posed nonlinear boundary value problems. We show how a suitable deep neural network architecture can be constructed and trained to satisfy the associated differential operators and the boundary conditions of the nonlinear problem. To show the accuracy of our method, we have tested the solutions arising from our method against known solutions of selected boundary value problems, e.g., comparison of the solution of Biharmonic equation arising from our convolutional neural network subject to the chosen boundary conditions with the corresponding analytical/numerical solutions. Furthermore, we demonstrate the accuracy, efficiency, and applicability of our method by solving the well known thin plate problem and the Navier-Stokes equation.

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.

A radialbasis function (RBF)–differentialquadrature (DQ) method is applied for the numerical solution of elliptic boundary value problems (BVPs) in three-dimensional axisymmetric domains. By appropriately selecting the collocation points, for any choice of RBF, the proposed discretization leads to linear systems in which the coefficient matrices possess block circulant structures. Matrix decomposition algorithms (MDAs) and fast Fourier transforms (FFTs) are employed for the efficient solution of these systems. Three types of BVPs are considered, namely ones governed by the Poisson equation, the inhomogeneous biharmonic equation, and the inhomogeneous Cauchy–Navier equations of elasticity. The high accuracy of the proposed technique as well as its ability to solve large-scale problems is demonstrated on several numerical examples.

In this paper, a constructive method is given for constructing the Green function of the Dirichlet problem for a biharmonic equation in a multidimensional ball. The need to study boundary value problems for elliptic equations is dictated by numerous practical applications in the theoretical study of the processes of hydrodynamics, electrostatics, mechanics, thermal conductivity, elasticity theory, and quantum physics. The distributions of the potential of the electrostatic field are described using the Poisson equation. When studying the vibrations of thin plates of small deflections, biharmonic equations arise. There are various ways to construct the Green Function of the Dirichlet problem for the Poisson equation. For many types of domains, it is constructed explicitly. And for the Neumann problem in multidimensional domains, the construction of the Green function is an open problem. For the ball, the Green function of the internal and external Neumann problem is constructed explicitly only for the two-dimensional and three-dimensional cases. Finding general correct boundary value problems for differential equations is always an urgent problem. The abstract theory of operator contraction and expansion originates from the work of John von Neumann, in which a method for constructing self-adjoint extensions of a symmetric operator was described and a theory of extension of symmetric operators with finite defect indices was developed in detail. Many problems for partial differential equations lead to operators with infinite defect indices. In the early 80s of the last century, M.O. Otelbaev and his students built an abstract theory that allows us to describe all correct constrictions of a certain maximum operator and separately - all correct extensions of a certain minimum operator, independently of each other, in terms of the inverse operator. In this paper, the correct boundary value problems for the biharmonic operator are described using the Green's function.

The error caused by inaccuracy in solving systems of equations by iterative methods is investigated. An upper estimate for an axially symmetric heat conduction equation is found for the error accumulated in several time steps. The estimate shows a linear dependence of the error on the threshold value of a criterion for limiting the number of iterations, a quadratic growth of the error depending on the number of points in space, and its independence of the number of steps in time. A computational experiment shows good agreement of the estimate with real errors at boundary and initial conditions of various types. A quadratic growth for Laplace’s equation of the error caused by an accuracy limitation in using an iterative method, depending on the number of points in space $$n$$ , is found empirically. A growth of $$n^{4}$$ for the similar error in the biharmonic equation is found.

In this paper, the localized Trefftz method (LTM) is proposed to accurately and efficiently solve two-dimensional boundary value problems, governed by Laplace and biharmonic equations, in complex domains. The LTM is formed by combining the classical indirect Trefftz method and the localization approach, so the LTM, free from mesh and numerical quadrature, has great potential for solving large-scale problems. For problems in multiply-connected domains, the solutions expressions in the proposed LTM is much simpler and more compact than that in the conventional indirect Trefftz method due to the localization concept and the overlapping subdomains. In the proposed LTM, both of the interior nodes and boundary nodes are required and the algebraic equation at each node, represents the satisfaction of governing equation or boundary condition, can be derived by implementing the Trefftz method at every subdomain. By enforcing the satisfaction of governing equations at every interior node and of boundary conditions at every boundary node, a sparse system of linear algebraic equations can be yielded. Then, the numerical solution of the proposed LTM can be efficiently obtained by solving the sparse system. Several numerical examples in simply-connected and multiply-connected domains are provided to demonstrate the accuracy and simplicity of the proposed LTM. Besides, the extremely-accurate solutions of the LTM are simultaneously demonstrated.

In this study, we develop an approximate formulation for a generalized form of the biharmonic problem based on pseudospectral meshless radial point interpolation (PSMRPI). The boundary conditions are considered as simply supported or clamped, with application to the theory of static analysis of thin-plates. The rigorous steps to analyze such problem are defining the high order derivatives, implementing multiple boundary conditions especially when the geometry of the domain of the problem is complex. In PSMRPI method the nodal points do not need to be regularly distributed and can even be quite arbitrary. It is easy to have high order derivatives of unknowns in terms of the values at nodal points by constructing operational matrices. Furthermore, it is observed that the multiple boundary conditions can be imposed by applying PSMRPI on nodal points near the boundaries of the domain. The main results on the generalized biharmonic problem are demonstrated by some examples to show the validity and trustworthiness of PSMRPI technique. Also, a comparison with the previously standard studied method for the biharmonic problem is done.

We establish a concentration-compactness principle for the Sobolev space W2,p(⋅)(Ω)∩W01,p(⋅)(Ω) that is a tool for overcoming the lack of compactness of the critical Sobolev imbedding. Using this r...

In this paper we study the existence and the multiplicity of nontrivial weak solutions for a fourth order variable exponent Kirchhoff type problem involving p(x)-biharmonic operator with changing sign weight and with no flux boundary condition. By using variational approach and the theory of variable exponent Sobolev spaces, we determine an interval of parameters for which this problem admits at least two nontrivial weak solutions.

In this paper, a posteriori error estimates for the Weak Galerkin finite element methods (WG-FEMs) for second order elliptic problems are derived in terms of an H 1 − equivalent energy norm. Corresponding estimators based on the helmholtz decomposition yield globally upper and locally lower bounds for the approximation errors of the WG-FEMs. Especially, the error analysis of our methods is proved to be valid for polygonal meshes (e.g., hybrid, polytopal non-convex meshes and those with hanging nodes) under general assumptions. In addition, the work can make adaptive WG-FEMs solving partial differential equations such as stokes equations and biharmonic equations on polygonal meshes possible. Finally, we verify the theoretical findings by a few numerical examples.

In this paper, we get the existence of two positive solutions for a fourth-order problem with Navier boundary condition. Our nonlinearity has a critical growth, and the method is a local minimum theorem obtained by Bonanno.

In this paper, we are interested in an inverse problem for the biharmonic equation to recover the Robin coefficients on a non-accessible part of the boundary in simply connected planar domain from a measured Riquier-Neumann data on the accessible part of the boundary. Our work is based on a system of nonlinear and ill-posed integral equations which is solved by Tikhonov regularization method to complete missing Cauchy data. The minimization problem will be considered to solve the inverse problem, and recover the Robin coefficients on a known boundary curve. We show the feasibility of this method by numerical reconstructions.

We introduced the Airy stress function in the non-Euclidean model of a continuous medium, for which the Saint-Venant compatibility condition for deformations is not satisfied. For this function, an inhomogeneous biharmonic equation is obtained whose right side is determined by the incompatibility of deformations. The internal stresses was shown to be composed of elastic stresses and stresses determined by the incompatibility of deformations. The theoretical results obtained are used to analyse experimental data on the study of residual stresses.

It is prominent that the Galerkin-type representation plays a dominant role in probing various challenges of mathematical physics, continuum mechanics and occupies an important place in the field of partial differential equations (PDEs). Thus, the contemporary analysis of different boundary value problems (BVPs) in thermoelasticity theory commonly begins by analyzing the Galerkin-type representation of the field equations in terms of elementary functions (harmonic, biharmonic, and metaharmonic, etc). This work is aimed at formulating the representation of a Galerkin-type solution by means of elementary functions for the recently developed Moore–Gibson–Thompson (MGT) thermoelasticity theory. The MGT theory is a generalized form of the Lord–Shulman (LS) model as well as of the Green–Naghdi (GN) thermoelastic model. Here, we establish a theorem and derive the Galerkin-type solution for the basic governing equations under this theory. Later, the Galerkin representation of a system of equations for steady oscillations is derived. Based on this representation, we finally establish the general solution (GS) for the system of homogeneous equations of stable oscillation, neglecting the extrinsic body force and extrinsic heat supply.

In this paper we consider a severely ill posed problem associated with a two dimensional homogeneous biharmonic equation. By perturbing the original problem and using a two parameters regularization method, we get a stable solution which converges to the solution of the considered problem. Under some priori bound assumptions, different errors estimates for the regularized solution are obtained. These last ones depend on the choice of the space of the exact solution. To show the effectiveness of the proposed regularization method some numerical results are given.

The biharmonic problem is a fourth order partial differential equation and thus requires two boundary conditions, and not just one like in the Laplace case, where the notion of Dirichlet and Neuman...

International audience; We introduce a residual-based a posteriori error estimator for a novel hp-version interior penalty discontinuous Galerkin method for the biharmonic problem in two and three dimensions. We prove that the error estimate provides an upper bound and a local lower bound on the error and that the lower bound is robust to the local mesh size but not the local polynomial degree. The suboptimality in terms of the polynomial degree is fully explicit and grows at most algebraically. Our analysis does not require the existence of a C1-conforming piecewise polynomial space and is instead based on an elliptic reconstruction of the discrete solution to the H2 space and a generalised Helmholtz decomposition of the error. This is the first hp-version error estimator for the biharmonic problem in two and three dimensions. The practical behaviour of the estimator is investigated through numerical examples in two and three dimensions.

In this article, we study the existence of at least three distinct weak solutions for nonlocal elliptic problems involving $ p(x) $-biharmonic operator. The results are obtained by means of variational methods. We also provide an example in order to illustrate our main abstract results. We extend and improve some recent results.

We prove the existence, positivity, simplicity, uniqueness up to nonnegative eigenfunctions, and isolation of the principle eigenvalue of a biharmonic system. We also provide the extension of our results to a related system.

This paper is concerned with the existence of an eigenvalue for a p(x)-biharmonic Kirchhoff problem with Navier boundary condition. Under some suitable conditions, we establish that any λ > 0 is an eigenvalue . The proofs combine variational methods with energy estimates. The main results of this paper improve and generalize the previous one introduced by Kefi and Radulescu (Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 29 (2018), 439-463).

The purpose of this paper is to study the existence of sign-changing solution to the following fourth-order equation:$$ \Delta ^{2}u- \biggl(a+ b \int _{\mathbb{R}^{N}} \vert \nabla u \vert ^{2}\,dx \biggr) \Delta u+V(x)u=K(x)f(u) \quad\text{in } \mathbb{R}^{N}, $$Δ2u−(a+b∫RN|∇u|2dx)Δu+V(x)u=K(x)f(u)in RN,where$5\leq N\leq 7$5≤N≤7,$\Delta ^{2}$Δ2denotes the biharmonic operator,$K(x), V(x)$K(x),V(x)are positive continuous functions which vanish at infinity, and$f(u)$f(u)is only a continuous function. We prove that the equation has a least energy sign-changing solution by the minimization argument on the sign-changing Nehari manifold. If, additionally,fis an odd function, we obtain that equation has infinitely many nontrivial solutions.

We consider a two dimensional molecular beam epitaxy equation with biharmonic dissipation. We give an improved gradient bound, which removes the logarithmic dependence on the dissipation coefficient previously shown in [1] .

From the eigenvalue problem theory, we see that the convergence rate of the biharmonic eigenvalues obtained by the mixed method in I. Bab u ˇ ska and J. Osborn, [Eigenvalue Problems, Handbook of Nu...

An initial-boundary value problem of the form ${D}_{t}^{\alpha } u+{\varDelta }^{2}u-c{\varDelta } u =f$ is considered, where ${D}_{t}^{\alpha }$ is a Caputo temporal derivative of order α ∈ (0,1) and c is a nonnegative constant. The spatial domain ${\varOmega } \subset \mathbb {R}^{d}$ for some d ∈{1,2,3}, with Ω bounded and convex. The boundary conditions are u = Δu = 0 on ∂Ω. A priori bounds on the solution are established, given sufficient regularity and compatibility of the data; typical solutions have a weak singularity at the initial time t = 0. The problem is rewritten as a system of two second-order differential equations, then discretised using standard finite elements in space together with the L1 discretisation of ${D}_{t}^{\alpha }$ on a graded temporal mesh. The numerical method computes approximations ${u_{h}^{n}}$ and ${{p}_{h}^{n}}$ of u(⋅,tn) and Δu(⋅,tn) at each time level tn. The stability of the method (i.e. a priori bounds on $\|{{u}_{h}^{n}}\|_{L^{2}({\varOmega })}$ and $\|{p_{h}^{n}}\|_{L^{2}({\varOmega })}$ ) is established by means of a new discrete Gronwall inequality that is α-robust, i.e. remains valid as α → 1−. Error bounds on $\|u(\cdot , t_{n}) - {u_{h}^{n}}\|_{L^{2}({\varOmega })}$ and $\|{\varDelta } u(\cdot , t_{n}) - {{p}_{h}^{n}}\|_{L^{2}({\varOmega })}$ are then derived; these bounds are of optimal order in the spatial and temporal mesh parameters for each fixed value of α, and they are α-robust if one considers α → 1−.

A well-known conjecture of Bang Yen-Chen says that the only biharmonic Euclidean submanifolds are minimal ones. In this paper, we consider an extended condition (namely, $L_1$-biharmonicity) on non-degenerate timelike hypersurfaces of the pseudo-Euclidean space $E_1^4$. A Lorentzian hypersurface $x: M_1^3\rightarrow\E_1^4$ is called $L_1$-biharmonic if it satisfies the condition $L_1^2x=0$, where $L_1$ is the linearized operator associated to the first variation of 2-th mean curvature vector field on $M_1^3$. According to the multiplicities of principal curvatures, the $L_1$-extension of Chen's conjecture is affirmed for Lorentzian hypersurfaces with constant ordinary mean curvature in pseudo-Euclidean space $E_1^4$. Additionally, we show that there is no proper $L_1$-biharmonic $L_1$-finite type connected orientable Lorentzian hypersurface in $E_1^4$.

A new $$C^0$$ nonconforming quasi three degree element with 13 freedoms is introduced to solve biharmonic problem. The given finite element space consists of piecewise polynomial space $$P_3$$ and some bubble functions. Different from non- $$C^0$$ nonconforming scheme, a smoother discrete solution can be obtained by this method. Compared with the existed 16 freedoms finite element method, this scheme uses less freedoms. As the finite elements are not affine equivalent each other, the associated interpolating error estimation is technically proved by introducing another affine finite elements. With this space to solve biharmonic problem, the convergence analysis is demonstrated between true solution and discrete solution. Under a stronger hypothesis that true solution $$u\in H_0^2(\Omega )\cap H^4(\Omega )$$ , the scheme is of linear order convergence by the measurement of discrete norm $$\Vert \cdot \Vert _h$$ . Some numerical results are included to further illustrate the convergence analysis.

The aim of this paper is to derive pointwise global and local best approximation type error estimates for biharmonic problems using the C 0 C^0 interior penalty method. The analysis uses the technique of dyadic decompositions of the domain, which is assumed to be a convex polygon. The proofs require local energy estimates and new pointwise Green’s function estimates for the continuous problem which has independent interest.

This paper is devoted to the study of the homogeneous Dirichlet problem for a singular nonlinear equation which involves the p(·)-biharmonic operator and a Hardy-type term that depend on the solution and with a parameter λ. By using a variational approach and min-max argument based on Ljusternik-Schnirelmann theory on C 1-manifolds [13], we prove that the considered problem admits at least one nondecreasing sequence of positive eigencurves with a characterization of the principal curve μ 1(λ) and also show that, the smallest curve μ 1(λ) is positive for all 0 ≤ λ < C H , with C H is the optimal constant of Hardy type inequality.

In this paper, three meshless methods are proposed to solve boundary value problems. The special purpose Trefftz function method, the method of fundamental solutions and the symmetry method of fundamental solutions are compared. We considered six numerical examples for harmonic and biharmonic problems as a flow through the cylindrical fibers arranged in a regular square array, deflection of a uniformly loaded square plate and torsion of a bar. The boundary value problems defined for a repeating element of the domain are solved by use of the collocation technique. The accuracy and stability of compared methods are investigated. Root mean square error on the boundary and condition number of the matrix are presented as functions of a number of collocation points.

In this paper, we study the existence and multiplicity of solutions for a class of equations involving a nonlocal term and the biharmonic operator with critical exponent. By new techniques, multiplicity results are established together with variational method. It is worth noting that the method of this paper is obviously different from the existing literature.

We present an effective numerical method for the recovery of biharmonic functions in a disk from continuous boundary values of these functions and of their normal derivatives using wavelets that are harmonic in the disk and interpolating on its boundary on dyadic rational grids. The expansions of solutions of boundary value problems into cumbersome interpolation series in the wavelet basis are replaced by sequences of their partial sums that are compactly presentable in the subspace bases of the corresponding multiresolution analysis (MRA) of Hardy spaces $$h_{\infty}(K)$$ of functions harmonic in the disk. Effective estimates are obtained for the approximation of solutions by partial sums of any order in terms of the best approximation of the boundary functions by trigonometric polynomials of a slightly smaller order. As a result, to provide the required accuracy of the representation of the unknown biharmonic functions, one can choose in advance the scaling parameter of the corresponding MRA subspace such that the interpolation projection to this space defines a simple analytic representation of the corresponding partial sums of interpolation series in terms of appropriate compressions and shifts of the scaling functions, skipping complicated iterative procedures for the numerical construction of the coefficients of expansion of the boundary functions into series in interpolating wavelets. We write solutions using interpolating and interpolating-orthogonal wavelets based on modified Meyer wavelets; the latter are convenient to apply if the boundary values of the boundary value problem are given approximately, for example, are found experimentally. In this case, one can employ the usual, well-known procedures of discrete orthogonal wavelet transformations for the analysis and refinement (correction) of the boundary values.

The purpose of the present paper is to study the existence of solutions for the following nonhomogeneous singular Kirchhoff problem involving the $p(x)$ -biharmonic operator: $$\begin{aligned} \left \{ \textstyle\begin{array}{l} M(t)\Big(\Delta ^{2}_{p(x)}u+a(x)|u|^{p(x)-2}u\Big) =g(x)u^{-\gamma (x)} \mp \lambda f(x,u),\quad \mbox{in }\Omega , \\ \Delta u=u=0, \quad \mbox{on }\partial \Omega , \end{array}\displaystyle \right . \end{aligned}$$ where $\Omega \subset {\mathbb{R}}^{N}$ , $(N\geq 3)$ be a bounded domain with $C^{2}$ boundary, $\lambda $ is a positive parameter, $\gamma : \overline{\Omega }\longrightarrow (0,1)$ be a continuous function, $p\in C(\overline{\Omega })$ with $\displaystyle 1< p^{-}:=\inf _{x\in \Omega }p(x)\leq p^{+}:=\sup _{x \in \Omega }p(x)

We formulate a well-posedness and approximation theory for a class of generalised saddle point problems with a specific form of constraints. In this way we develop an approach to a class of fourth order elliptic partial differential equations with point constraints using the idea of splitting into coupled second order equations. An approach is formulated using a penalty method to impose the constraints. Our main motivation is to treat certain fourth order equations involving the biharmonic operator and point Dirichlet constraints for example arising in the modelling of biomembranes on curved and flat surfaces but the approach may be applied more generally. The theory for well-posedness and approximation is presented in an abstract setting. Several examples are described together with some numerical experiments.

In this paper, we propose a multigrid method for the solution of the biharmonic problem. Isogeometric Analysis (IGA) is considered in order to easily obtain H 2 -conforming discretizations of the bilaplacian equation, which are difficult to get by means of standard finite element methods. Typically, the design of solvers for isogeometric discretizations that are robust with respect to the polynomial degree is a challenging task. Here, we achieve such robustness by using multiplicative Schwarz methods as smoothers within the multigrid algorithm. The design of the proposed solver is also supported by a local Fourier analysis (LFA), which allows us to choose appropriately the size of the block in the smoother depending on the polynomial degree of the discretization. The robustness and efficiency of the proposed multigrid method is demonstrated through numerical experiments in one- and two-dimensional cases.

There are three criteria when we model with phase-field equations. The first one is the evolution of phase separation. The second one is energy dissipation law. The last one is whether we have the law of conservation of mass or not. If we minimize the Ginzburg–Landau free-energy functional, the kinetics of phase separation inherently follows. However, the other properties such as energy dissipation and mass conservation are more difficult to model. Hitherto, there have been a few proposals to conserve mass using the Allen–Cahn (AC) equation, but we do not have the energy dissipation law in these models when considered in light of local effects of mass-correction. For the reasons, certain researchers still prefer to use the Cahn–Hilliard equation, e.g. biharmonic equation instead of using the Allen–Cahn equation when we consider a mass conserving case. This paper is concerned with numerical solutions of the AC equation in the mass-conserving space, where the solution is corrected at each iteration for mass conservation. We introduce two different methods for solving the conservative AC equation with periodic boundary conditions. In this work, we have the AC equation in common, but use two different mass correction operators. Note that both of mass correction operators satisfy the discrete energy dissipation property. First, we project the solution of the AC equation to the mass conserving space, and then recover the linear convex splitting scheme for the conventional mass-conserving equation. The linear convex splitting scheme is well-known but the convergence of numerical solutions is elaborately studied in terms of gradient-projection. Next, we propose the other energy-minimizing method on the mass-conserving space. It translates the solution of the AC equation to the mass conserving space via an energy-dissipation operator. The dissipation of discrete energy functional for the proposed method is also established. As far as the author knows, no report has been presented for the numerical scheme having discrete energy dissipation property of the mass-conserving AC equation without using the nonlocal and constant multiplier. To verify the common properties of mass-projection and proposed methods numerically, we simulate the evolutions for phase separation, dissipation of energy, and mass conservation. We moreover consider the short and long time stages of evolutions to see the difference between two methods. These several tests are provided for its applications of the proposed method for modeling natural phenomena.