Journal articles: 'Partial differential equations – Elliptic equations and systems – Higher-order elliptic equations'

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Tokibetov, Zh A., N. E. Bashar, and А. К. Pirmanova. "THE CAUCHY-DIRICHLET PROBLEM FOR A SYSTEM OF FIRST-ORDER EQUATIONS." BULLETIN Series of Physics & Mathematical Sciences 72, no. 4 (December 29, 2020): 68–72. http://dx.doi.org/10.51889/2020-4.1728-7901.10.

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For a single second-order elliptic partial differential equation with sufficiently smooth coefficients, all classical boundary value problems that are correct for the Laplace equations are Fredholm. The formulation of classical boundary value problems for the laplace equation is dictated by physical applications. The simplest of the boundary value problems for the Laplace equation is the Dirichlet problem, which is reduced to the problem of the field of charges distributed on a certain surface. The Dirichlet problem for partial differential equations in space is usually called the Cauchy-Dirichlet problem. This work dedicated to systems of first-order partial differential equations of elliptic and hyperbolic types consisting of four equations with three unknown variables. An explicit solution of the CauchyDirichlet problem is constructed using the method of an exponential – differential operator. Giving a very simple example of the co-solution of the Cauchy problem for a second-order differential equation and the Cauchy problem for systems of first-order hyperbolic differential equations.

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Song, Mingliang, and Shuyuan Mei. "Existence of Three Solutions for Nonlinear Operator Equations and Applications to Second-Order Differential Equations." Journal of Function Spaces 2021 (January 28, 2021): 1–17. http://dx.doi.org/10.1155/2021/6668037.

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The existence of three solutions for nonlinear operator equations is established via index theory for linear self-adjoint operator equations, critical point reduction method, and three critical points theorems obtained by Brezis-Nirenberg, Ricceri, and Averna-Bonanno. Applying the results to second-order Hamiltonian systems satisfying generalized periodic boundary conditions or Sturm-Liouville boundary conditions and elliptic partial differential equations satisfying Dirichlet boundary value conditions, we obtain some new theorems concerning the existence of three solutions.

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Wei, Lin. "Some second-order systems of partial differential equations of composite type." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 106, no. 1-2 (1987): 73–88. http://dx.doi.org/10.1017/s0308210500018217.

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SynopsisThe Cauchy problem and the Dirichlet-Cauchy type problem of some second-order systems of partial differential equations of composite type of two unknown functions are investigated. Such systems possess some of the characteristics not only of elliptic but also of hyperbolic systems in the same domain. Representations of the solutions are found for the upper half plane. To this end, the composite systems are reduced to the canonical form by means of successive applications of three kinds of linear transformations. Function theoretic methods are used to obtain representation formulae. Furthermore, some composite systems of 2m-unknown function are also considered.

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Li, Xiaoou, Jingchen Liu, and Shun Xu. "A multilevel approach towards unbiased sampling of random elliptic partial differential equations." Advances in Applied Probability 50, no. 4 (November 29, 2018): 1007–31. http://dx.doi.org/10.1017/apr.2018.49.

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Abstract Partial differential equations are powerful tools for used to characterizing various physical systems. In practice, measurement errors are often present and probability models are employed to account for such uncertainties. In this paper we present a Monte Carlo scheme that yields unbiased estimators for expectations of random elliptic partial differential equations. This algorithm combines a multilevel Monte Carlo method (Giles (2008)) and a randomization scheme proposed by Rhee and Glynn (2012), (2013). Furthermore, to obtain an estimator with both finite variance and finite expected computational cost, we employ higher-order approximations.

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Cosner, Chris, and Philip W. Schaefer. "Sign-definite solutions in some linear elliptic systems." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 111, no. 3-4 (1989): 347–58. http://dx.doi.org/10.1017/s030821050001862x.

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SynopsisWe consider a weakly coupled set of two partial differential equations where the coupling matrix has variable elements and the principal part of each equation is the same uniformly elliptic operator. Weobtain necessary conditions that the system of equations can be decoupled. By decoupling the system and using a positivity lemma due to Hess and Kato, we determine the algebraic sign of the solution components. This work extends recent results of de Figueiredo and Mitidieri. Further, one can use these results to determine the sign of the solution to certain fourth order elliptic boundary value problems.

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Sowa, Artur. "Modeling a Quantum Hall System via Elliptic Equations." Advances in Mathematical Physics 2009 (2009): 1–9. http://dx.doi.org/10.1155/2009/514081.

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Quantum Hall systems are a suitable theme for a case study in the general area of nanotechnology. In particular, it is a good framework to search for universal principles relevant to nanosystem modeling and nanosystem-specific signal processing. Recently, we have been able to construct a partial differential equations-based model of a quantum Hall system, which consists of the Schrödinger equation supplemented with a special-type nonlinear feedback loop. This result stems from a novel theoretical approach, which in particular brings to the fore the notion of quantum information. Here we undertake to modify the original model by substituting the dynamics based on the Dirac operator. This leads to a model that consists of a system of three nonlinearly coupled first-order elliptic equations in the plane.

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Ogiwara, Toshiko, and Hiroshi Matano. "Stability analysis in order-preserving systems in the presence of symmetry." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 129, no. 2 (1999): 395–438. http://dx.doi.org/10.1017/s0308210500021429.

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Given an equation with a certain symmetry, such as symmetry with respect to rotation or translation, one of the most fundamental questions to ask is whether or not the symmetry of the equation is inherited by its solutions. We first discuss this question in a general framework of order-preserving dynamical systems under a group action and establish a theory concerning symmetry or monotonicity properties of stable equilibrium points. We then apply this general theory to nonlinear partial differential equations. Among other things, we prove the rotational symmetry of solutions for a class of nonlinear elliptic equations and the monotonicity of travelling waves of some nonlinear diffusion equations. We also discuss the stability of stationary or periodic solutions for equations of surface motion.

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Shakhmurov, Veli B. "Maximal regular boundary value problems in Banach-valued function spaces and applications." International Journal of Mathematics and Mathematical Sciences 2006 (2006): 1–26. http://dx.doi.org/10.1155/ijmms/2006/92134.

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The nonlocal boundary value problems for differential operator equations of second order with dependent coefficients are studied. The principal parts of the differential operators generated by these problems are non-selfadjoint. Several conditions for the maximal regularity and the Fredholmness in Banach-valuedLp-spaces of these problems are given. By using these results, the maximal regularity of parabolic nonlocal initial boundary value problems is shown. In applications, the nonlocal boundary value problems for quasi elliptic partial differential equations, nonlocal initial boundary value problems for parabolic equations, and their systems on cylindrical domain are studied.

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TSAI, LONG-YI, and S. T. WU. "EXISTENCE OF SOLUTIONS FOR ELLIPTIC INTEGRO-DIFFERENTIAL SYSTEMS." Tamkang Journal of Mathematics 25, no. 1 (March 1, 1995): 61–70. http://dx.doi.org/10.5556/j.tkjm.25.1994.4426.

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In this paper the existence of the solution for elliptic integro-differential systems are discussed. Those systems are motivated by certain physical processes such as in epidemics, predator-prey dynamics and the others. We extend the method of mixed monotony to second order elliptic partial integro-differential equations. By assuming the existence of a satellite $f$ of the give function $\Phi$, we prove the existence of solutions by using fixed point theory. Moreover, we provide the modified method of mixed monotony to construct two monotone sequences which converge uniformly to the solution. We also give sufficient conditions for the existence of $f$ and obtain the construction of upper and lower solutions in some applications.

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Köster, M., and S. Turek. "The Influence of Higher Order FEM Discretisations on Multigrid Convergence." Computational Methods in Applied Mathematics 6, no. 2 (2006): 221–32. http://dx.doi.org/10.2478/cmam-2006-0011.

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AbstractQuadratic and even higher order finite elements are interesting candidates for the numerical solution of partial differential equations (PDEs) due to their improved approximation properties in comparison to linear approaches. The systems of equations that arise from the discretisation of the underlying (elliptic) PDEs are often solved by iterative solvers like preconditioned Krylow-space methods, while multigrid solvers are still rarely used – which might be caused by the high effort that is associated with the realisation of the necessary data structures as well as smoothing and intergrid transfer operators. In this note, we discuss the numerical analysis of quadratic conforming finite elements in a multigrid solver. Using the “correct” grid transfer operators in conjunction with a quadratic finite element approximation allows to formulate an improved approximation property which enhances the (asymptotic) behaviour of multigrid: If m denotes the number of smoothing steps, the convergence rates behave asymptotically like O(1/m2) in contrast to O(1/m) for linear FEM.

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Evans, Lawrence C. "The perturbed test function method for viscosity solutions of nonlinear PDE." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 111, no. 3-4 (1989): 359–75. http://dx.doi.org/10.1017/s0308210500018631.

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SynopsisThe method of viscosity solutions for nonlinear partial differential equations (PDEs) justifies passages to limits by in effect using the maximum principle to convert to the corresponding limit problem for smooth test functions. We describe in this paper a “perturbed test function” device, which entails various modifications of the test functions by lower order correctors. Applications include homogenisation for quasilinear elliptic PDEs and approximation of quasilinear parabolic PDEs by systems of Hamilton-Jacobi equations.

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Cywiak-Códova, D., G. Gutiérrez-Juárez, and And M. Cywiak-Garbarcewicz. "Spectral generalized function method for solving homogeneous partial differential equations with constant coefficients." Revista Mexicana de Física E 17, no. 1 Jan-Jun (January 28, 2020): 11. http://dx.doi.org/10.31349/revmexfise.17.11.

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A method based on a generalized function in Fourier space gives analytical solutions to homogeneous partial differential equations with constant coefficients of any order in any number of dimensions. The method exploits well-known properties of the Dirac delta, reducing the differential mathematical problem into the factorization of an algebraic expression that finally has to be integrated. In particular, the method was utilized to solve the most general homogeneous second order partial differential equation in Cartesian coordinates, finding a general solution for non-parabolic partial differential equations, which can be seen as a generalization of d'Alambert solution. We found that the traditional classification, i.e., parabolic, hyperbolic and elliptic, is not necessary reducing the classification to only parabolic and non-parabolic cases. We put special attention for parabolic partial differential equations, analyzing the general 1D homogeneous solution of the Photoacoustic and Photothermal equations in the frequency and time domain. Finally, we also used the method to solve Helmholtz equation in cylindrical coordinates, showing that it can be used in other coordinates systems.

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Kaewta, Supaporn, Sekson Sirisubtawee, and Nattawut Khansai. "Explicit Exact Solutions of the (2+1)-Dimensional Integro-Differential Jaulent–Miodek Evolution Equation Using the Reliable Methods." International Journal of Mathematics and Mathematical Sciences 2020 (April 27, 2020): 1–19. http://dx.doi.org/10.1155/2020/2916395.

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In this article, we utilize the G′/G2-expansion method and the Jacobi elliptic equation method to analytically solve the (2 + 1)-dimensional integro-differential Jaulent–Miodek equation for exact solutions. The equation is shortly called the Jaulent–Miodek equation, which was first derived by Jaulent and Miodek and associated with energy-dependent Schrödinger potentials (Jaulent and Miodek, 1976; Jaulent, 1976). The equation is converted into a fourth order partial differential equation using a transformation. After applying a traveling wave transformation to the resulting partial differential equation, we obtain an ordinary differential equation which is the main equation to which the both schemes are applied. As a first step, the two methods give us distinguish systems of algebraic equations. The first method provides exact traveling wave solutions including the logarithmic function solutions of trigonometric functions, hyperbolic functions, and polynomial functions. The second approach provides the Jacobi elliptic function solutions depending upon their modulus values. Some of the obtained solutions are graphically characterized by the distinct physical structures such as singular periodic traveling wave solutions and peakons. A comparison between our results and the ones obtained from the previous literature is given. Obtaining the exact solutions of the equation shows the simplicity, efficiency, and reliability of the used methods, which can be applied to other nonlinear partial differential equations taking place in mathematical physics.

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CHKADUA, O., S. E. MIKHAILOV, and D. NATROSHVILI. "ANALYSIS OF DIRECT SEGREGATED BOUNDARY-DOMAIN INTEGRAL EQUATIONS FOR VARIABLE-COEFFICIENT MIXED BVPs IN EXTERIOR DOMAINS." Analysis and Applications 11, no. 04 (June 18, 2013): 1350006. http://dx.doi.org/10.1142/s0219530513500061.

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Direct segregated systems of boundary-domain integral equations are formulated for the mixed (Dirichlet–Neumann) boundary value problems for a scalar second-order divergent elliptic partial differential equation with a variable coefficient in an exterior three-dimensional domain. The boundary-domain integral equation system equivalence to the original boundary value problems and the Fredholm properties and invertibility of the corresponding boundary-domain integral operators are analyzed in weighted Sobolev spaces suitable for infinite domains. This analysis is based on the corresponding properties of the BVPs in weighted Sobolev spaces that are proved as well.

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Yanushauskas, A. "Well-posedness of the Dirichlet problem and homotopy classification of elliptic systems of second-order partial differential equations." Banach Center Publications 19, no. 1 (1987): 371–81. http://dx.doi.org/10.4064/-19-1-371-381.

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Kopylov, A. P. "Stability in theC 1-norm of sheaves of solutions to elliptic systems of second-order linear partial differential equations." Siberian Mathematical Journal 39, no. 6 (December 1998): 1125–39. http://dx.doi.org/10.1007/bf02674124.

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Santo, Niccolò Dal, Simone Deparis, and Andrea Manzoni. "A numerical investigation of multi space reduced basis preconditioners for parametrized elliptic advection-diffusion equations." Communications in Applied and Industrial Mathematics 8, no. 1 (December 20, 2017): 282–97. http://dx.doi.org/10.1515/caim-2017-0015.

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AbstractWe analyze the numerical performance of a preconditioning technique recently proposed in [1] for the efficient solution of parametrized linear systems arising from the finite element (FE) discretization of parameterdependent elliptic partial differential equations (PDEs). In order to exploit the parametric dependence of the PDE, the proposed preconditioner takes advantage of the reduced basis (RB) method within the preconditioned iterative solver employed to solve the linear system, and combines a RB solver, playing the role of coarse component, with a traditional fine grid (such as Additive Schwarz or block Jacobi) preconditioner. A sequence of RB spaces is required to handle the approximation of the error-residual equation at each step of the iterative method at hand, whence the name of Multi Space Reduced Basis (MSRB) method. In this paper, a numerical investigation of the proposed technique is carried on in the case of a Richardson iterative method, and then extended to the flexible GMRES method, in order to solve parameterized advection-diffusion problems. Particular attention is payed to the impact of anisotropic diffusion coefficients and (possibly dominant) transport terms on the proposed preconditioner, by carrying out detailed comparisons with the current state of the art algebraic multigrid preconditioners.

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Ferrari, Paola, Ryma Imene Rahla, Cristina Tablino-Possio, Skander Belhaj, and Stefano Serra-Capizzano. "Multigrid for Q k Finite Element Matrices Using a (Block) Toeplitz Symbol Approach." Mathematics 8, no. 1 (December 18, 2019): 5. http://dx.doi.org/10.3390/math8010005.

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In the present paper, we consider multigrid strategies for the resolution of linear systems arising from the Q k Finite Elements approximation of one- and higher-dimensional elliptic partial differential equations with Dirichlet boundary conditions and where the operator is div − a ( x ) ∇ · , with a continuous and positive over Ω ¯ , Ω being an open and bounded subset of R 2 . While the analysis is performed in one dimension, the numerics are carried out also in higher dimension d ≥ 2 , showing an optimal behavior in terms of the dependency on the matrix size and a substantial robustness with respect to the dimensionality d and to the polynomial degree k.

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COHEN, ALBERT, RONALD DEVORE, and CHRISTOPH SCHWAB. "ANALYTIC REGULARITY AND POLYNOMIAL APPROXIMATION OF PARAMETRIC AND STOCHASTIC ELLIPTIC PDE'S." Analysis and Applications 09, no. 01 (January 2011): 11–47. http://dx.doi.org/10.1142/s0219530511001728.

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Parametric partial differential equations are commonly used to model physical systems. They also arise when Wiener chaos expansions are used as an alternative to Monte Carlo when solving stochastic elliptic problems. This paper considers a model class of second order, linear, parametric, elliptic PDE's in a bounded domain D with coefficients depending on possibly countably many parameters. It shows that the dependence of the solution on the parameters in the diffusion coefficient is analytically smooth. This analyticity is then exploited to prove that under very weak assumptions on the diffusion coefficients, the entire family of solutions to such equations can be simultaneously approximated by multivariate polynomials (in the parameters) with coefficients taking values in the Hilbert space [Formula: see text] of weak solutions of the elliptic problem with a controlled number of terms N. The convergence rate in terms of N does not depend on the number of parameters in V which may be countable, therefore breaking the curse of dimensionality. The discretization of the coefficients from a family of continuous, piecewise linear finite element functions in D is shown to yield finite dimensional approximations whose convergence rate in terms of the overall number N dof of degrees of freedom is the minimum of the convergence rates afforded by the best N-term sequence approximations in the parameter space and the rate of finite element approximations in D for a single instance of the parametric problem.

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BONILLA, L. L., C. J. PÉREZ VICENTE, F. RITORT, and J. SOLER. "EXACT SOLUTIONS AND DYNAMICS OF GLOBALLY COUPLED OSCILLATORS." Mathematical Models and Methods in Applied Sciences 16, no. 12 (December 2006): 1919–59. http://dx.doi.org/10.1142/s0218202506001765.

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We analyze mean-field models of synchronization of phase oscillators with singular couplings and subject to external random forces. They are related to the Kuramoto–Sakaguchi model. Their probability densities satisfy local partial differential equations similar to the porous medium, Burgers and extended Burgers systems depending on the degree of singularity of the coupling. We show that porous medium oscillators (the most singularly coupled) do not synchronize and that (transient) synchronization is possible only at zero temperature for Burgers oscillators. The extended Burgers oscillators have a nonlocal coupling first introduced by Daido and they may synchronize at any temperature. Exact expressions for their synchronized phases and for Daido's order function are given in terms of elliptic functions.

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Papaefthymiou, E. S., D. T. Papageorgiou, and G. A. Pavliotis. "Nonlinear interfacial dynamics in stratified multilayer channel flows." Journal of Fluid Mechanics 734 (October 8, 2013): 114–43. http://dx.doi.org/10.1017/jfm.2013.443.

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AbstractThe dynamics of viscous immiscible pressure-driven multilayer flows in channels are investigated using a combination of modelling, analysis and numerical computations. More specifically, the particular system of three stratified layers with two internal fluid–fluid interfaces is considered in detail in order to identify the nonlinear mechanisms involved due to multiple fluid surface interactions. The approach adopted is analytical/asymptotic and is valid for interfacial waves that are long compared with the channel height or individual undisturbed liquid layer thicknesses. This leads to a coupled system of fully nonlinear partial differential equations of Benney type that contain a small slenderness parameter that cannot be scaled out of the problem. This system is in turn used to develop a consistent coupled system of weakly nonlinear evolution equations, and it is shown that this is possible only if the underlying base-flow and fluid parameters satisfy certain conditions that enable a synchronous Galilean transformation to be performed at leading order. Two distinct canonical cases (all terms in the equations are of the same order) are identified in the absence and presence of inertia, respectively. The resulting systems incorporate all of the active physical mechanisms at Reynolds numbers that are not large, namely, nonlinearities, inertia-induced instabilities (at non-zero Reynolds number) and surface tension stabilization of sufficiently short waves. The coupled system supports several instabilities that are not found in single long-wave equations including, transitional instabilities due to a change of type of the flux nonlinearity from hyperbolic to elliptic, kinematic instabilities due to the presence of complex eigenvalues in the linearized advection matrix leading to a resonance between the interfaces, and the possibility of long-wave instabilities induced by an interaction between the flux function of the system and the surface tension terms. All of these instabilities are followed into the nonlinear regime by carrying out extensive numerical simulations using spectral methods on periodic domains. It is established that instabilities leading to coherent structures in the form of nonlinear travelling waves are possible even at zero Reynolds number, in contrast to single interface (two-layer) systems; in addition, even in parameter regimes where the flow is linearly stable, the coupling of the flux functions and their hyperbolic–elliptic transitions lead to coherent structures for initial disturbances above a threshold value. When inertia is present an additional short-wave instability enters and the systems become general coupled Kuramoto–Sivashinsky-type equations. Extensive numerical experiments indicate a rich landscape of dynamical behaviour including nonlinear travelling waves, time-periodic travelling states and chaotic dynamics. It is also established that it is possible to regularize the chaotic dynamics into travelling wave pulses by enhancing the inertialess instabilities through the advective terms. Such phenomena may be of importance in mixing, mass and heat-transfer applications.

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Best Mckay, Maricela, Brittany A. Erickson, and Jeremy E. Kozdon. "A computational method for earthquake cycles within anisotropic media." Geophysical Journal International 219, no. 2 (July 15, 2019): 816–33. http://dx.doi.org/10.1093/gji/ggz320.

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SUMMARY We present a numerical method for the simulation of earthquake cycles on a 1-D fault interface embedded in a 2-D homogeneous, anisotropic elastic solid. The fault is governed by an experimentally motivated friction law known as rate-and-state friction which furnishes a set of ordinary differential equations which couple the interface to the surrounding volume. Time enters the problem through the evolution of the ordinary differential equations along the fault and provides boundary conditions for the volume, which is governed by quasi-static elasticity. We develop a time-stepping method which accounts for the interface/volume coupling and requires solving an elliptic partial differential equation for the volume response at each time step. The 2-D volume is discretized with a second-order accurate finite difference method satisfying the summation-by-parts property, with boundary and fault interface conditions enforced weakly. This framework leads to a provably stable semi-discretization. To mimic slow tectonic loading, the remote side-boundaries are displaced at a slow rate, which eventually leads to earthquake nucleation at the fault. Time stepping is based on an adaptive, fourth-order Runge–Kutta method and captures the highly varying timescales present. The method is verified with convergence tests for both the orthotropic and fully anisotropic cases. An initial parameter study reveals regions of parameter space where the systems experience a bifurcation from period one to period two behaviour. Additionally, we find that anisotropy influences the recurrence interval between earthquakes, as well as the emergence of aseismic transients and the nucleation zone size and depth of earthquakes.

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Malanchuk, Yevhenii, Viktor Moshynskyi, Valerii Korniienko, and Zinovii Malanchuk. "Modeling the process of hydromechanical amber extraction." E3S Web of Conferences 60 (2018): 00005. http://dx.doi.org/10.1051/e3sconf/20186000005.

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The paper represents a process of hydromechanical amber extraction modeling to obtain input data and substantiate operation parameters of mining equipment to develop the improved hydromechanical technique of amber extraction. Intensification of amber mining process is possible when sandy deposit is saturated by water and air. Moreover, mechanical impact by means of vibration is added. Amber displacement within sandy deposit is considered. The deposit is characterized by environmental resistance when influence factors act on the process of amber surfacing. Amber concentration distribution over a deposit surface involving determination of floating periods of different amber fractions in terms of different operation modes as well as computer experiment concerning the amber grades and its distribution over amber-bearing deposit involved the use of computer environment Matlab. Adequate mathematical model to solve one-dimensional boundary problems for systems of parabolic and elliptic differential equations within partial first-order derivatives on one spatial variable and time has been developed. The model describes accurately the behaviour of different amber fractions within amber-bearing deposit in terms of vibration effect as well as water and air supply.

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Kadioglu, Samet Y. "A Second-Order IMEX Method for Multi-Phase Flow Problems." International Journal of Computational Methods 14, no. 05 (November 22, 2016): 1750056. http://dx.doi.org/10.1142/s0219876217500566.

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We present a fully second order IMplicit/EXplicit (IMEX) time integration technique for solving incompressible multi-phase flow problems. A typical incompressible multi-phase flow model consists of the Navier–Stokes equations plus an interface dynamics equation (e.g., the level set equation). Our IMEX strategy is applied to such a model in the following manner. The hyperbolic terms of the Navier–Stokes equations together with the interface dynamics equation are solved explicitly (Explicit Block) making use of the well-understood explicit numerical schemes [Leveque, R. J. [1998] Finite Volume Methods for Hyperbolic Problems, “Texts in Applied Mathematics”, (Cambridge University Press); Thomas, J. W. [1999] Numerical Partial Differential Equations II (Conservation Laws and Elliptic Equations), “Texts in Applied Mathematics” (Springer-Verlag, New York)]. On the other hand, the nonhyperbolic (stiff) parts of the flow equations are solved implicitly (Implicit Block) within the framework of the Jacobian-Free Newton Krylov (JFNK) method [Knoll, D. A. and Keyes, D. E. [2004] Jacobian-free Newton Krylov methods: A survey of approaches and applications. J. Comput. Phys. 193, 357–397; Saad, Y. [2003] Iterative Methods for Sparse Linear Systems (Siam); Kelley, C. T. [2003] Solving Nonlinear Equations with Newton’s Method (Siam)]. In our algorithm implementation, the explicit block is embedded in the implicit block in a way that it is always part of the nonlinear function evaluation. In this way, there exists a continuous interaction between the implicit and explicit algorithm blocks meaning that the improved solutions (in terms of time accuracy) at each nonlinear iteration are immediately felt by the explicit block and the improved explicit solutions are readily available to form the next set of nonlinear residuals. This continuous interaction between the two algorithm blocks results in an implicitly balanced algorithm in that all nonlinearities due to coupling of different time terms are converged with the desired numerical time accuracy. In other words, we obtain a self-consistent IMEX method that eliminates the possible order reductions in time convergence that is quite common in certain types of nonlinearly coupled systems. We remark that an incompressible multi-phase flow model can be a highly nonlinearly coupled system with the involvement of very stiff surface tension source terms. These kinds of flow problems are difficult to tackle numerically. In other words, highly nonlinear surface terms may remain unconverged leading to time inaccuracies or time order reductions to the first order even though the overall numerical scheme is designed as high order (second-order or higher) [Sussman, M. and Ohta, M. [2009] A stable and efficient method for treating surface tension in incompressible two-phase flow, SIAM J. Sci. Comput. 31(4), 2447–2471; Zheng, W., Zhu, B., Kim, B. and Fedkiw, R. [2015] A new incompressibility discretization for a hybrid particle MAC grid representation with surface tension, J. Comput. Phys. 280, 96–142]. These and few more issues are addressed in this paper. We have numerically tested our newly proposed scheme by solving several multi-phase flow settings such as an air bubble rising in water, a Rayleigh–Taylor instability problem that is initiated by placing a heavy fluid on top of a lighter one, and a droplet problem in which a water droplet hits the pool of water. Our numerical results show that we have achieved the second-order time accuracy without any order reductions. Moreover, the interfaces between the fluids are captured reasonably well.

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Muzhinji, Kizito, and Stanford Shateyi. "A Robust Approximation of the Schur Complement Preconditioner for an Efficient Numerical Solution of the Elliptic Optimal Control Problems." Computation 8, no. 3 (July 27, 2020): 68. http://dx.doi.org/10.3390/computation8030068.

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In this paper, we consider the numerical solution of the optimal control problems of the elliptic partial differential equation. Numerically tackling these problems using the finite element method produces a large block coupled algebraic system of equations of saddle point form. These systems are of large dimension, block, sparse, indefinite and ill conditioned. The solution of such systems is a major computational task and poses a greater challenge for iterative techniques. Thus they require specialised methods which involve some preconditioning strategies. The preconditioned solvers must have nice convergence properties independent of the changes in discretisation and problem parameters. Most well known preconditioned solvers converge independently of mesh size but not for the decreasing regularisation parameter. This work proposes and extends the work for the formulation of preconditioners which results in the optimal performances of the iterative solvers independent of both the decreasing mesh size and the regulation parameter. In this paper we solve the indefinite system using the preconditioned minimum residual method. The main task in this work was to analyse the 3 × 3 block diagonal preconditioner that is based on the approximation of the Schur complement form obtained from the matrix system. The eigenvalue distribution of both the proposed Schur complement approximate and the preconditioned system will be investigated since the clustering of eigenvalues points to the effectiveness of the preconditioner in accelerating an iterative solver. This is done in order to create fast, efficient solvers for such problems. Numerical experiments demonstrate the effectiveness and performance of the proposed approximation compared to the other approximations and demonstrate that it can be used in practice. The numerical experiments confirm the effectiveness of the proposed preconditioner. The solver used is robust and optimal with respect to the changes in both mesh size and the regularisation parameter.

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Aksoy, Ü., and A. O. çelebi. "Schwarz problem for higher-order complex elliptic partial differential equations." Integral Transforms and Special Functions 19, no. 6 (June 2008): 413–28. http://dx.doi.org/10.1080/10652460801933645.

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McCORMICK, STEPHEN F., and ULRICH RÜDE. "ON LOCAL REFINEMENT HIGHER ORDER METHODS FOR ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS." International Journal of High Speed Computing 02, no. 04 (December 1990): 311–34. http://dx.doi.org/10.1142/s0129053390000194.

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Wang, Yuan-Ming. "Higher-order Lidstone boundary value problems for elliptic partial differential equations." Journal of Mathematical Analysis and Applications 308, no. 1 (August 2005): 314–33. http://dx.doi.org/10.1016/j.jmaa.2005.01.019.

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Holtby, Derek W. "Higher-order estimates for fully nonlinear difference equations. I." Proceedings of the Edinburgh Mathematical Society 43, no. 3 (October 2000): 485–510. http://dx.doi.org/10.1017/s0013091500021155.

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AbstractThe purpose of this work is to establish a priori C2, α estimates for mesh function solutions of nonlinear positive difference equations in fully nonlinear form on a uniform mesh, where the fully nonlinear finite-difference operator ℱh is concave in the second-order variables. The estimate is an analogue of the corresponding estimate for solutions of concave fully nonlinear elliptic partial differential equations. We deal here with the special case that the operator does not depend explicitly upon the independent variables. We do this by discretizing the approach of Evans for fully nonlinear elliptic partial differential equations using the discrete linear theory of Kuo and Trudinger. The result in this special case forms the basis for a more general result in part II. We also derive the discrete interpolation inequalities needed to obtain estimates for the interior C2, α semi-norm in terms of the C0 norm.

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Holtby, Derek W. "HIGHER-ORDER ESTIMATES FOR FULLY NONLINEAR DIFFERENCE EQUATIONS. II." Proceedings of the Edinburgh Mathematical Society 44, no. 1 (February 2001): 87–102. http://dx.doi.org/10.1017/s0013091598000200.

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AbstractThe purpose of this work is to establish a priori $C^{2,\alpha}$ estimates for mesh function solutions of nonlinear difference equations of positive type in fully nonlinear form on a uniform mesh, where the fully nonlinear finite difference operator $\F$ is concave in the second-order variables. The estimate is an analogue of the corresponding estimate for solutions of concave fully nonlinear elliptic partial differential equations. We use the results for the special case that the operator does not depend explicitly upon the independent variables (the so-called frozen case) established in part I to approach the general case of explicit dependence upon the independent variables. We make our approach for the diagonal case via a discretization of the approach of Safonov for fully nonlinear elliptic partial differential equations using the discrete linear theory of Kuo and Trudinger and an especially agreeable mesh function interpolant provided by Kunkle. We generalize to non-diagonal operators using an idea which, to the author’s knowledge, is novel. In this paper we establish the desired Hölder estimate in the large, that is, on the entire mesh $n$-plane. In a subsequent paper a truly interior estimate will be established in a mesh $n$-box.AMS 2000 Mathematics subject classification: Primary 35J60; 35J15; 39A12. Secondary 39A70; 39A10; 65N06; 65N22; 65N12

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Contreras, Martha. "Resonance for Quasilinear Elliptic Higher Order Partial Differential Equations at the First Eigenvalue." Rocky Mountain Journal of Mathematics 28, no. 2 (June 1998): 417–44. http://dx.doi.org/10.1216/rmjm/1181071779.

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Achatz, S. "Higher Order Sparse Grid Methods for Elliptic Partial Differential Equations with Variable Coefficients." Computing 71, no. 1 (August 1, 2003): 1–15. http://dx.doi.org/10.1007/s00607-003-0012-8.

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Rathish Kumar, B. V., and Gopal Priyadarshi. "Wavelet Galerkin method for fourth-order multi-dimensional elliptic partial differential equations." International Journal of Wavelets, Multiresolution and Information Processing 16, no. 05 (September 2018): 1850045. http://dx.doi.org/10.1142/s0219691318500455.

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We describe a wavelet Galerkin method for numerical solutions of fourth-order linear and nonlinear partial differential equations (PDEs) in 2D and 3D based on the use of Daubechies compactly supported wavelets. Two-term connection coefficients have been used to compute higher-order derivatives accurately and economically. Localization and orthogonality properties of wavelets make the global matrix sparse. In particular, these properties reduce the computational cost significantly. Linear system of equations obtained from discretized equations have been solved using GMRES iterative solver. Quasi-linearization technique has been effectively used to handle nonlinear terms arising in nonlinear biharmonic equation. To reduce the computational cost of our method, we have proposed an efficient compression algorithm. Error and stability estimates have been derived. Accuracy of the proposed method is demonstrated through various examples.

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Nagaraja, K. V., V. Kesavulu Naidu, and P. G. Siddheshwar. "Optimal Subparametric Finite Elements for Elliptic Partial Differential Equations Using Higher-Order Curved Triangular Elements." International Journal for Computational Methods in Engineering Science and Mechanics 15, no. 2 (March 4, 2014): 83–100. http://dx.doi.org/10.1080/15502287.2013.870256.

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Contreras, M. "Double resonance for quasilinear elliptic higher order partial differential equations between the first and second eigenvalues." Nonlinear Analysis: Theory, Methods & Applications 25, no. 12 (December 1995): 1257–81. http://dx.doi.org/10.1016/0362-546x(94)00246-e.

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LI, WU, and TIAN YOU FAN. "STUDY ON ELASTIC ANALYSIS OF CRACK PROBLEM OF TWO-DIMENSIONAL DECAGONAL QUASICRYSTALS OF POINT GROUP 10, $\overline {10}$." Modern Physics Letters B 23, no. 16 (June 30, 2009): 1989–99. http://dx.doi.org/10.1142/s0217984909020151.

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By introducing a stress potential function, we transform the plane elasticity equations of two-dimensional quasicrystals of point group 10, [Formula: see text] to a partial differential equation. And then we use the complex variable function method for classical elasticity theory to that of the quasicrystals. As an example, a decagonal quasicrystal in which there is an arc is subjected to a uniform pressure p in the elliptic notch of the decagonal quasicrystal is considered. With the help of conformal mapping, we obtain the exact solution for the elliptic notch problem of quasicrystals. The work indicates that the stress potential and complex variable function methods are very useful for solving the complicated boundary value problems of higher order partial differential equations which originate from quasicrystal elasticity.

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NEUBERGER, JOHN M., NÁNDOR SIEBEN, and JAMES W. SWIFT. "AUTOMATED BIFURCATION ANALYSIS FOR NONLINEAR ELLIPTIC PARTIAL DIFFERENCE EQUATIONS ON GRAPHS." International Journal of Bifurcation and Chaos 19, no. 08 (August 2009): 2531–36. http://dx.doi.org/10.1142/s0218127409024293.

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We seek solutions u ∈ ℝn to the semilinear elliptic partial difference equation -Lu + fs(u) = 0, where L is the matrix corresponding to the Laplacian operator on a graph G and fs is a one-parameter family of nonlinear functions. This article combines the ideas introduced by the authors in two papers: (a) Nonlinear elliptic partial difference equations on graphs (J. Experimental Mathematics, 2006), which introduces analytical and numerical techniques for solving such equations, and (b) Symmetry and automated branch following for a semilinear elliptic PDE on a fractal region (SIAM J. Dynamical Systems, 2006), wherein we present some of our recent advances concerning symmetry, bifurcation and automation for PDE. We apply the symmetry analysis found in the SIAM paper to arbitrary graphs in order to obtain better initial guesses for Newton's method, create informative graphics, and better understand the role of symmetry in the underlying variational structure. We use two modified implementations of the gradient Newton–Galerkin algorithm (GNGA, Neuberger and Swift) to follow bifurcation branches in a robust way. By handling difficulties that arise when encountering accidental degeneracies and higher-dimensional critical eigenspaces, we can find many solutions of many symmetry types to the discrete nonlinear system. We present a selection of experimental results which demonstrate our algorithm's capability to automatically generate bifurcation diagrams and solution graphics starting with only an edgelist of a graph. We highlight interesting symmetry and variational phenomena.

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Rehman, Mehvish Fazal Ur, Yongyi Gu, and Wenjun Yuan. "Exact Analytical Solutions of Generalized Fifth-Order KdV Equation by the Extended Complex Method." Journal of Function Spaces 2021 (May 7, 2021): 1–9. http://dx.doi.org/10.1155/2021/5549288.

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The recently introduced technique, namely, the extended complex method, is used to explore exact solutions for the generalized fifth-order KdV equation. Appropriately, the rational, periodic, and elliptic function solutions are obtained by this technique. The 3D graphs explain the different physical phenomena to the exact solutions of this equation. This idea specifies that the extended complex method can acquire exact solutions of several differential equations in engineering. These results reveal that the extended complex method can be directly and easily used to solve further higher-order nonlinear partial differential equations (NLPDEs). All computer simulations are constructed by maple packages.

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Li, Biao, Yong Chen, and Yu-Qi Li. "A Generalized Sub-Equation Expansion Method and Some Analytical Solutions to the Inhomogeneous Higher-Order Nonlinear Schrödinger Equation." Zeitschrift für Naturforschung A 63, no. 12 (December 1, 2008): 763–77. http://dx.doi.org/10.1515/zna-2008-1204.

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On the basis of symbolic computation a generalized sub-equation expansion method is presented for constructing some exact analytical solutions of nonlinear partial differential equations. To illustrate the validity of the method, we investigate the exact analytical solutions of the inhomogeneous high-order nonlinear Schrödinger equation (IHNLSE) including not only the group velocity dispersion, self-phase-modulation, but also various high-order effects, such as the third-order dispersion, self-steepening and self-frequency shift. As a result, a broad class of exact analytical solutions of the IHNLSE are obtained. From our results, many previous solutions of some nonlinear Schrödinger-type equations can be recovered by means of suitable selections of the arbitrary functions and arbitrary constants. With the aid of computer simulation, the abundant structure of bright and dark solitary wave solutions, combined-type solitary wave solutions, dispersion-managed solitary wave solutions, Jacobi elliptic function solutions and Weierstrass elliptic function solutions are shown by some figures.

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Akinyemi, Lanre, Najib Ullah, Yasir Akbar, Mir Sajjad Hashemi, Arzu Akbulut, and Hadi Rezazadeh. "Explicit solutions to nonlinear Chen–Lee–Liu equation." Modern Physics Letters B 35, no. 25 (August 12, 2021): 2150438. http://dx.doi.org/10.1142/s0217984921504388.

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In this work, a generalized [Formula: see text]-expansion method has been used for solving the nonlinear Chen–Lee–Liu equation. This method is a more common, general, and powerful mathematical algorithm for finding the exact solutions of nonlinear partial differential equations (NPDEs), where [Formula: see text] follows the Jacobi elliptic equation [Formula: see text], and we let [Formula: see text] be a fourth-order polynomial. Many new exact solutions such as the hyperbolic, rational, and trigonometric solutions with different parameters in terms of the Jacobi elliptic functions are obtained. The distinct solutions obtained in this paper clearly explain the importance of some physical structures in the field of nonlinear phenomena. Also, this method deals very well with higher-order nonlinear equations in the field of science. The numerical results described in the plots were obtained by using Maple.

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Liu, Guojun, Wentao Ma, Hailong Ma, and Lin Zhu. "A multiple-scale higher order polynomial collocation method for 2D and 3D elliptic partial differential equations with variable coefficients." Applied Mathematics and Computation 331 (August 2018): 430–44. http://dx.doi.org/10.1016/j.amc.2018.03.021.

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Heinz, Hans-Peter. "Lacunary bifurcation for operator equations and nonlinear boundary value problems on ℝN." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 118, no. 3-4 (1991): 237–70. http://dx.doi.org/10.1017/s0308210500029073.

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SynopsisWe consider nonlinear eigenvalue problems of the form Lu + F(u) = λu in a real Hilbert space, where L is a positive self-adjoint linear operator and F is a nonlinearity vanishing to higher order at u = 0. We suppose that there are gaps in the essential spectrum of L and use critical point theory for strongly indefinite functionals to derive conditions for the existence of non-zero solutions for λ belonging to such a gap, and for the bifurcation of such solutions from the line of trivial solutions at the boundary points of a gap. The abstract results are applied to the L2-theory of semilinear elliptic partial differential equations on ℝN. We obtain existence results for the general case and bifurcation results for nonlinear perturbations of the periodic Schrödinger equation.

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Ern, Alexandre, and Pietro Zanotti. "A quasi-optimal variant of the hybrid high-order method for elliptic partial differential equations with H−1 loads." IMA Journal of Numerical Analysis 40, no. 4 (January 15, 2020): 2163–88. http://dx.doi.org/10.1093/imanum/drz057.

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Abstract Hybrid high-order (HHO) methods for elliptic diffusion problems have been originally formulated for loads in the Lebesgue space $L^2(\varOmega )$. In this paper we devise and analyse a variant thereof, which is defined for any load in the dual Sobolev space $H^{-1}(\varOmega )$. The main feature of the present variant is that its $H^1$-norm error can be bounded only in terms of the $H^1$-norm best error in a space of broken polynomials. We establish this estimate with the help of recent results on the quasi-optimality of nonconforming methods. We prove also an improved error bound in the $L^2$-norm by duality. Compared to previous works on quasi-optimal nonconforming methods the main novelties are that HHO methods handle pairs of unknowns and not a single function and, more crucially, that these methods employ a reconstruction that is one polynomial degree higher than the discrete unknowns. The proposed modification affects only the formulation of the discrete right-hand side. This is obtained by properly mapping discrete test functions into $H^1_0(\varOmega )$.

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Heydari, Zohreh, Gholamreza Shobeyri, and Seyed Hossein Ghoreishi Najafabadi. "Accuracy analysis of different higher-order Laplacian models of incompressible SPH method." Engineering Computations 37, no. 1 (July 19, 2019): 181–202. http://dx.doi.org/10.1108/ec-02-2019-0057.

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Purpose This paper aims to examine the accuracy of several higher-order incompressible smoothed particle hydrodynamics (ISPH) Laplacian models and compared with the classic model (Shao and Lo, 2003). Design/methodology/approach The numerical errors in solving two-dimensional elliptic partial differential equations using the Laplacian models are investigated for regular and highly irregular node distributions over a unit square computational domain. Findings The numerical results show that one of the Laplacian models, which is newly developed by one of the authors (Shobeyri, 2019) can get the smallest errors for various used node distributions. Originality/value The newly proposed model is formulated by the hybrid of the standard ISPH Laplacian model combined with Taylor expansion and moving least squares method. The superiority of the proposed model is significant when multi-resolution irregular node distributions commonly seen in adaptive refinement strategies used to save computational cost are applied.

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Seadawy, Aly R., Mujahid Iqbal, and Dianchen Lu. "Analytical methods via bright–dark solitons and solitary wave solutions of the higher-order nonlinear Schrödinger equation with fourth-order dispersion." Modern Physics Letters B 33, no. 35 (December 16, 2019): 1950443. http://dx.doi.org/10.1142/s0217984919504438.

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In this research work, we investigated the higher-order nonlinear Schrödinger equation (NLSE) with fourth-order dispersion, self-steepening, nonlinearity, nonlinear dispersive terms and cubic-quintic terms which is described as the propagation of ultra-short pulses in fiber optics. We apply the modification form of extended auxiliary equation mapping method to find the new exact and solitary wave solutions of higher-order NLSE. As a result, new solutions are obtained in the form of solitons, kink–anti-kink type solitons, bright–dark solitons, traveling wave, trigonometric functions, elliptic functions and periodic solitary wave solutions. These new different types of solutions show the power and fruitfulness of this new method and also show two- and three-dimensional graphically with the help of computer software Mathematica. These new solutions have many applications in the field of physics and other branches of physical sciences. We can also solve other higher-order nonlinear partial differential equations (NPDEs) involved in mathematical physics and other various branches of physical sciences with this new technique.

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Dick, Josef, Robert N. Gantner, Quoc T. Le Gia, and Christoph Schwab. "Multilevel higher-order quasi-Monte Carlo Bayesian estimation." Mathematical Models and Methods in Applied Sciences 27, no. 05 (April 19, 2017): 953–95. http://dx.doi.org/10.1142/s021820251750021x.

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We propose and analyze deterministic multilevel (ML) approximations for Bayesian inversion of operator equations with uncertain distributed parameters, subject to additive Gaussian measurement data. The algorithms use a ML approach based on deterministic, higher-order quasi-Monte Carlo (HoQMC) quadrature for approximating the high-dimensional expectations, which arise in the Bayesian estimators, and a Petrov–Galerkin (PG) method for approximating the solution to the underlying partial differential equation (PDE). This extends the previous single-level (SL) approach from [J. Dick, R. N. Gantner, Q. T. Le Gia and Ch. Schwab, Higher order quasi-Monte Carlo integration for Bayesian estimation, Report 2016-13, Seminar for Applied Mathematics, ETH Zürich (in review)]. Compared to the SL approach, the present convergence analysis of the ML method requires stronger assumptions on holomorphy and regularity of the countably-parametric uncertainty-to-observation maps of the forward problem. As in the SL case and in the affine-parametric case analyzed in [J. Dick, F. Y. Kuo, Q. T. Le Gia and Ch. Schwab, Multi-level higher order QMC Galerkin discretization for affine parametric operator equations, SIAM J. Numer. Anal. 54 (2016) 2541–2568], we obtain sufficient conditions which allow us to achieve arbitrarily high, algebraic convergence rates in terms of work, which are independent of the dimension of the parameter space. The convergence rates are limited only by the spatial regularity of the forward problem, the discretization order achieved by the PG discretization, and by the sparsity of the uncertainty parametrization. We provide detailed numerical experiments for linear elliptic problems in two space dimensions, with [Formula: see text] parameters characterizing the uncertain input, confirming the theory and showing that the ML HoQMC algorithms can outperform, in terms of error versus computational work, both multilevel Monte Carlo (MLMC) methods and SL HoQMC methods, provided the parametric solution maps of the forward problems afford sufficient smoothness and sparsity of the high-dimensional parameter spaces.

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Ingham, D. B., and I. Pop. "Natural convection about a heated horizontal cylinder in a porous medium." Journal of Fluid Mechanics 184 (November 1987): 157–81. http://dx.doi.org/10.1017/s0022112087002842.

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The natural convection from a heated circular cylinder in an unbounded region of porous medium is investigated for the full range of Rayleigh numbers. At small Rayleigh numbers a qualitative solution is obtained and at large Rayleigh numbers the second-order boundary-layer solution is found that takes into account the first-order plume solution. In order to find the solution at finite Rayleigh numbers the two governing coupled, nonlinear, elliptic partial differential equations are expressed in finite-difference form using a specialized technique which is second-order accurate everywhere. Further, methods are devised which deal with the plume and infinity boundary conditions. Although numerical results are presented for Rayleigh numbers up to 400 solutions of the finite-difference equations can be obtained for higher values of the Rayleigh numbers but in these cases the mesh size used is too large to adequately deal with the developing boundary-layer on the cylinder and the plume.The numerical results show how the theories at both low and high Rayleigh numbers are approached. The plume solution which develops with increasing Rayleigh number agrees with that predicted by the theory presented using the boundary-layer approximation. No separation of the flow at the top of the cylinder is found and there are no indications that it will appear at higher values of the Rayleigh number. The results presented here give reasonable agreement with the existing experimental results for Rayleigh numbers of order unity. However as the Rayleigh number increases to order 102 there is a large discrepancy between the theoretical and experimental results and this is because at these higher values of the Rayleigh number the Darcy approximation has been violated in the experimental results. This indicates the severe limitations of some of the existing theories which use boundary-layer analyses and the Darcy approximation for flows in a porous medium. The application of Darcy's law requires that the size of the pores be much smaller than the scale of the bulk flow and inertial and thermal lengthscales.

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Youssef, Maha, and Roland Pulch. "Poly-Sinc Solution of Stochastic Elliptic Differential Equations." Journal of Scientific Computing 87, no. 3 (April 30, 2021). http://dx.doi.org/10.1007/s10915-021-01498-9.

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AbstractIn this paper, we introduce a numerical solution of a stochastic partial differential equation (SPDE) of elliptic type using polynomial chaos along side with polynomial approximation at Sinc points. These Sinc points are defined by a conformal map and when mixed with the polynomial interpolation, it yields an accurate approximation. The first step to solve SPDE is to use stochastic Galerkin method in conjunction with polynomial chaos, which implies a system of deterministic partial differential equations to be solved. The main difficulty is the higher dimensionality of the resulting system of partial differential equations. The idea here is to solve this system using a small number of collocation points in space. This collocation technique is called Poly-Sinc and is used for the first time to solve high-dimensional systems of partial differential equations. Two examples are presented, mainly using Legendre polynomials for stochastic variables. These examples illustrate that we require to sample at few points to get a representation of a model that is sufficiently accurate.

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Chkadua, Otar, Sergey Mikhailov, and David Natroshvili. "Localized boundary-domain singular integral equations of the Robin type problem for self-adjoint second-order strongly elliptic PDE systems." Georgian Mathematical Journal, November 12, 2020. http://dx.doi.org/10.1515/gmj-2020-2082.

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AbstractThe paper deals with the three-dimensional Robin type boundary-value problem (BVP) for a second-order strongly elliptic system of partial differential equations in the divergence form with variable coefficients. The problem is studied by the localized parametrix based potential method. By using Green’s representation formula and properties of the localized layer and volume potentials, the BVP under consideration is reduced to the a system of localized boundary-domain singular integral equations (LBDSIE). The equivalence between the original boundary value problem and the corresponding LBDSIE system is established. The matrix operator generated by the LBDSIE system belongs to the Boutet de Monvel algebra. With the help of the Vishik–Eskin theory based on the Wiener–Hopf factorization method, the Fredholm properties of the corresponding localized boundary-domain singular integral operator are investigated and its invertibility in appropriate function spaces is proved.

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Bulíček, Miroslav, Erika Maringová, and Josef Málek. "On nonlinear problems of parabolic type with implicit constitutive equations involving flux." Mathematical Models and Methods in Applied Sciences, August 25, 2021, 1–52. http://dx.doi.org/10.1142/s0218202521500457.

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We study systems of nonlinear partial differential equations of parabolic type, in which the elliptic operator is replaced by the first-order divergence operator acting on a flux function, which is related to the spatial gradient of the unknown through an additional implicit equation. This setting, broad enough in terms of applications, significantly expands the paradigm of nonlinear parabolic problems. Formulating four conditions concerning the form of the implicit equation, we first show that these conditions describe a maximal monotone [Formula: see text]-coercive graph. We then establish the global-in-time and large-data existence of a (weak) solution and its uniqueness. To this end, we adopt and significantly generalize Minty’s method of monotone mappings. A unified theory, containing several novel tools, is developed in a way to be tractable from the point of view of numerical approximations.

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Journal articles on the topic 'Partial differential equations – Elliptic equations and systems – Higher-order elliptic equations'

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