In this paper we consider a class of fourth order nonlinear integro-differential equations with Navier boundary conditions. By the reduction of the problem to operator equation we establish the existence and uniqueness of solution and construct a numerical method for solving it. We prove that the method is of second order accuracy and obtain an estimate for total error. Some examples demonstrate the validity of the obtained theoretical results and the efficiency of the numerical method.
In this paper, we consider a boundary value problem (BVP) for a fourth order nonlinear functional integro-differential equation. We establish the existence and uniqueness of solution and construct a numerical method for solving it. We prove that the method is of second order accuracy and obtain an estimate for the total error. Some examples demonstrate the validity of the obtained theoretical results and the efficiency of the numerical method.
In this paper we consider a boundary value problem for fully fourth order nonlinear functional differential equation which contains all lower derivatives of proportional delay arguments. By the reduction of the problem to operator equation for the right hand side nonlinear function we establish the existence and uniqueness of solution and construct iterative methods on both continuous and discrete levels for solving it. We obtain the total error estimate for the discrete iterative solution. Many examples demonstrate the validity of the obtained theoretical results and the efficiency of the num…
This paper establishes and analyzes a second-order accurate numerical scheme for the nonlinear partial integrodifferential equation with a weakly singular kernel. In the time direction, we apply the Crank-Nicolson method for the time derivative, and the product-integration (PI) rule is employed to deal with Riemann-Liouville fractional integral. From which, the non-uniform meshes are utilized to compensate for the singular behavior of the exact solution at $t=0$ so that our method can reach second-order convergence for time. In order to formulate a fully discrete implicit difference scheme, we…
In this work, we consider a boundary value problem for nonlinear triharmonic equation. Due to the reduction of nonlinear boundary value problems to operator equation for nonlinear terms we establish the existence, uniqueness and positivity of solution. More importantly, we design an iterative method at both continuous and discrete level for numerical solution of the problem. An analysis of actual total error of the obtained discrete solution is made. Some examples demonstrate the applicability of the theoretical results on qualitative aspects and the efficiency of the iterative method.
This paper presents an efficient spectral method for solving the fractional Fredholm integro-differential equations. The non-smoothness of the solutions to such problems leads to the performance of spectral methods based on the classical polynomials such as Chebyshev, Legendre, Laguerre, etc, with a low order of convergence. For this reason, the development of classic numerical methods to solve such problems becomes a challenging issue. Since the non-smooth solutions have the same asymptotic behavior with polynomials of fractional powers, therefore, fractional basis functions are the best cand…
Existence results and numerical solution for the Dirichlet problem for fully fourth order nonlinear equation
Center for Informatics and Computing, Vietnam Academy of Science and Technology(VAST)（越南科学技术院信息计算中心）；College of Sciences, Thainguyen University（太原大学理学院）
Iterative solution of a nonlinear static beam equation
Georgian Technical University（格鲁吉亚理工大学）；A.Razmadze Mathematical Institute, I. Javakhishvili Tbilisi State University（I·贾瓦希什维利第比利斯国立大学A·拉兹马泽数学研究所）；Faculty of Exact and Natural Sciences, I. Javakhishvili Tbilisi State University（I·贾瓦希什维利第比利斯国立大学精确与自然科学学院）；I. Vekua Institute of Applied Mathematics, I. Javakhishvili Tbilisi State University（I·贾瓦希什维利第比利斯国立大学I·韦库阿应用数学研究所）；V. Komarovi Tbilisi Physics and Mathematics №199 Public School（V·科马里夫第比利斯物理与数学第199公立学校）
Fast multilevel solvers for a class of discrete fourth order parabolic problems
Pacific Northwest National Laboratory（太平洋西北国家实验室）；Southwest Jiaotong University（西南交通大学）；Tufts University（塔夫茨大学）；University of California, Irvine（加州大学尔湾分校）；University of Maryland（马里兰大学）；Pennsylvania State University（宾夕法尼亚州立大学）
A least-squares method for solving the hyperbolic Monge-Ampère equation with transport boundary condition is introduced. The method relies on an iterative procedure for the gradient of the solution, the so-called mapping. By formulating error functionals for the interior domain, the boundary, both separately and as linear combination, three minimization problems are solved iteratively to compute the mapping. After convergence, a fourth minimization problem, to compute the solution of the Monge-Ampère equation, is solved. The approach is based on a least-squares method for the elliptic Monge-Am…
We focus here on a class of fourth-order parabolic equations that can be written as a system of second-order equations by introducing an auxiliary variable. We design a novel second-order fully discrete mixed finite element method to approximate these equations. In our approach, we propose new techniques using the second-order backward differentiation formula for the time derivative and a special technique for the approximation of nonlinear terms. The use of the proposed technique for nonlinear terms makes the developed numerical scheme efficient in terms of computational cost since the propos…
In this work, we consider the Dirichlet boundary value problem for nonlinear triharmonic equation. Due to the reduction of the nonlinear boundary value problem to operator equation for the nonlinear term and the unknown second normal derivative we design an iterative method at both continuous and discrete level for numerical solution of the problem. Some examples demonstrate that the numerical method is of fourth order convergence.
Iterative methods with certified convergence for the computation of Gauss--Jacobi quadratures are described. The methods do not require a priori estimations of the nodes to guarantee its fourth-order convergence. They are shown to be generally faster than previous methods and without practical restrictions on the range of the parameters. The evaluation of the nodes and weights of the quadrature is exclusively based on convergent processes which, together with the fourth order convergence of the fixed point method for computing the nodes, makes this an ideal approach for high accuracy computati…
The existence of numerical solutions to a fourth order singular boundary value problem arising in the theory of epitaxial growth is studied. An iterative numerical method is applied on a second order nonlinear singular boundary value problem which is the exact result of the reduction of this fourth order singular boundary value problem. It turns out that the existence or nonexistence of numerical solutions fully depends on the value of a parameter. We show that numerical solutions exist for small positive values of this parameter. For large positive values of the parameter, we find nonexistenc…
This paper analyzes the convergence of fixed-point iterations of the form u = f(u) and the properties of the inverse of the related pentadiagonal matrices, associated with the fourth-order nonlinear beam equation. This nonlinear problem is discretized using the finite difference method with the clamped-free and clamped-clamped boundary conditions in the one dimension. Explicit formulas for the inverse of the matrices and norms of the inverse are derived. In iterative process, the direct computation of inverse matrix allows to achieve an efficiency. Numerical results were provided.
Unless special conditions apply, the attempt to solve ill-conditioned systems of linear equations with standard numerical methods leads to uncontrollably high numerical error. Often, such systems arise from the discretization of operator equations with a large number of discrete variables. In this paper we show that the accuracy can be improved significantly if the equation is transformed before discretization, a process we call full operator preconditioning (FOP). It bears many similarities with traditional preconditioning for iterative methods but, crucially, transformations are applied at t…
The existence and nonexistence of stationary radial solutions to the elliptic partial differential equation arising in the molecular beam epitaxy are studied. The fourth-order radial equation is non-self adjoint and has no exact solutions. Also, it admits multiple solutions. Furthermore, solutions depend on the size of the parameter. We show that solutions exist for small positive values of this parameter. For large positive values of this parameter, we prove the nonexistence of solutions. We establish the qualitative properties of the solutions and provide bounds for the values of this parame…
We study a self-similar solution of the kinetic equation describing weak wave turbulence in Bose-Einstein condensates. This solution presumably corresponds to an asymptotic behavior of a spectrum evolving from a broad class of initial data, and it features a non-equilibrium finite-time condensation of the wave spectrum $n(ω)$ at the zero frequency $ω$. The self-similar solution is of the second kind, and it satisfies boundary conditions corresponding to a nonzero constant spectrum (with all its derivative being zero) at $ω=0$ and a power-law asymptotic $n(ω) \to ω^{-x}$ at $ω\to \infty \;\; x\…
Dynamical systems are pervasive in almost all engineering and scientific applications. Simulating such systems is computationally very intensive. Hence, Model Order Reduction (MOR) is used to reduce them to a lower dimension. Most of the MOR algorithms require solving large sparse sequences of linear systems. Since using direct methods for solving such systems does not scale well in time with respect to the increase in the input dimension, efficient preconditioned iterative methods are commonly used. In one of our previous works, we have shown substantial improvements by reusing preconditioner…
Enabling Local Neural Operators to perform Equation-Free System-Level Analysis
Hopkins Extreme Materials Institute, Johns Hopkins University（霍普金斯极端材料研究所，约翰霍普金斯大学）；Dept. of Chemical and Biomolecular Engineering, Johns Hopkins University（化学与生物分子工程系，约翰霍普金斯大学）；Dept. of Applied Mathematics and Statistics, Johns Hopkins University（应用数学与统计学系，约翰霍普金斯大学）；Dept. of Civil and Systems Engineering, Johns Hopkins University（土木与系统工程系，约翰霍普金斯大学）；Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”, Università degli Studi di Napoli Federico II（雷纳托·卡奇奥波利数学与应用系，那不勒斯费德里克二世大学）；School of Medicine’s Dept. of Urology, Johns Hopkins University（医学学院泌尿科系，约翰霍普金斯大学）
