Evolution Partial Differential Equations Research Articles

Complexification and integrability in multidimensions

The complexification of the independent variables of nonlinear integrable evolution partial differential equations (PDEs) in two space dimensions, like the celebrated Kadomtsev-Petviashvili and Davey-Stewartson (DS) equations, yields nonlinear integrable equations in genuine 4 + 2, namely, in four real space dimensions (x1, x2, y1, y2) and two real time dimensions (t1, t2), as opposed to two complex space dimensions and one complex time dimension. The associated initial value problem for such equations, namely, the problem where the dependent variables are specified for all space variables at t1 = t2 = 0, can be solved via a non-local d-bar formalism. Here, the details of this formalism for the 4 + 2 DS system are presented. Furthermore, the linearised version of the 3 + 1 reduction of the 4 + 2 DS system is discussed. The construction of the nonlinear 3 + 1 reduction remains open, in spite of the fact that multi-soliton solutions for the 3 + 1 DS system already exist.

Continuation methods for time-periodic travelling-wave solutions to evolution equations

A numerical continuation method is developed to follow time-periodic travelling-wave solutions of both local and non-local evolution partial differential equations (PDEs). It is found that the equation for the speed of the moving coordinate can be derived naturally from the governing equations together with a condition that breaks the translational symmetry. The derived system of equations allows one to follow the branch of travelling-wave solutions as well as solutions that are time-periodic in a frame of reference travelling at a constant speed. Finally, we show as an example the bifurcation and stability analysis of single and double-pulse waves in long-wave models of electrified falling films.

A personalized mathematical tool for neuro-oncology: A clinical case study

This work evaluates the predictive ability of a novel personalized computational tool for simulating the growth of brain tumours using the neuroimaging data collected during one clinical case study. The mathematical model consists of an evolutionary fourth-order partial differential equation with degenerate motility, in which the spreading dynamics of the multiphase tumour is coupled with a parabolic equation determining the diffusing oxygen within the brain. The model also includes a reaction term describing the effects of radiotherapy, that is simulated in accordance with the clinical schedule. We collect Magnetic Resonance (MRI) and Diffusion Tensor (DTI) imaging data for one patient at given times of key clinical interest, from the first diagnosis of a giant glioblastoma to its surgical removal and the subsequent radiation therapies. These neuroimaging data allow reconstructing the patient-specific brain geometry in a finite element virtual environment, that is used for simulating the tumour recurrence pattern after the surgical resection. In particular, we characterize the different brain tissues and the tumour location from MRI data, whilst we extrapolate the heterogeneous nutrient diffusion parameters and cellular motility from DTI data.The numerical results of the simulated tumour are found in good qualitative and quantitative agreement with the volume and the boundaries observed in MRI. Moreover, the simulations point out a consistent regression of the tumour mass in correspondence to the application of radiotherapy, with an average growth rate which is of the same order as the one calculated from the neuroimaging data. Remarkably, our results display the highest Jaccard index of the tumour region reported in the biomathematical literature. In conclusion, this work represents an important proof-of-concept of the ability of this mathematical framework to predict the tumour recurrence and its response to therapies in a patient-specific manner.

English

Advanced building design is a rather new interdisciplinary research branch, combining knowledge from physics, engineering, art and social science; its support from both theoretical and computational mathematics is needed. This paper shows an example of such collaboration, introducing a model problem of optimal heating in a low-energy house. Since all particular function values, needed for optimization are obtained as numerical solutions of an initial and boundary value problem for a sparse system of parabolic partial differential equations of evolution with at least two types of physically motivated nonlinearities, the usual gradient-based methods must be replaced by the downhill simplex Nelder-Mead approach or its quasi-gradient modifications. One example of the real low-energy house in Moravian Karst is demonstrated with references to other practical applications.

Nonlocal and Multipoint Boundary Value Problems for Linear Evolution Equations

AbstractWe derive the solution representation for a large class of nonlocal boundary value problems for linear evolution partial differential equations (PDE) with constant coefficients in one space variable. The prototypical example of such PDE is the heat equation, for which problems of this form model physical phenomena in chemistry and for which we formulate and prove a full result. We also consider the third‐order case, which is much less studied and has been shown by the authors to have very different structural properties in general. The nonlocal conditions we consider can be reformulated as multipoint conditions, and then an explicit representation for the solution of the problem is obtained by an application of the Fokas transform method. The analysis is carried out under the assumption that the problem being solved is well posed, i.e., it admits a unique solution. For the second‐order case, we also give criteria that guarantee well posedness.

Solving parameter estimation problems with discrete adjoint exponential integrators

The solution of inverse problems in a variational setting finds best estimates of the model parameters by minimizing a cost function that penalizes the mismatch between model outputs and observations. The gradients required by the numerical optimization process are computed using adjoint models. Exponential integrators are a promising family of time discretization schemes for evolutionary partial differential equations. In order to allow the use of these discretization schemes in the context of inverse problems, adjoints of exponential integrators are required. This work derives the discrete adjoint formulae for W-type exponential propagation iterative methods of Runge–Kutta type (EPIRK-W). These methods allow arbitrary approximations of the Jacobian while maintaining the overall accuracy of the forward integration. The use of Jacobian approximation matrices that do not depend on the model state avoids the complex calculation of Hessians in the discrete adjoint formulae. The adjoint code itself is generated efficiently via algorithmic differentiation and used to solve inverse problems with the Lorenz-96 model and a model from computational magnetics. Numerical results are encouraging and indicate the suitability of exponential integrators for this class of problems.

Algebraic aspects of evolution partial differential equation arising in the study of constant elasticity of variance model from financial mathematics

AbstractThe optimal investment-consumption problem under the constant elasticity of variance (CEV) model is investigated from the perspective of Lie group analysis. The Lie symmetry group of the evolution partial differential equation describing the CEV model is derived. The Lie point symmetries are then used to obtain an exact solution of the governing model satisfying a standard terminal condition. Finally, we construct conservation laws of the underlying equation using the general theorem on conservation laws.

Modeling solid-state dewetting of a single-crystal binary alloy thin films

Dewetting of a binary alloy thin film is studied using a continuum many-parameter model that accounts for the surface and bulk diffusion, the bulk phase separation, the surface segregation, and the particle formation. An analytical solution is found for the quasistatic equilibrium concentration of a surface-segregated atomic species. This solution is factored into the nonlinear and coupled evolution partial differential equations (PDEs) for the bulk composition and surface morphology. The stability of a planar film surface with respect to small perturbations of shape and composition is analyzed, revealing the dependence of the particle size on major physical parameters. The computations show various scenarios of the particle formation and the redistribution of the alloy components inside the particles and on their surface. In most situations, for the alloy film composed initially of 50% A and 50% B atoms, core-shell particles are formed, and they are located atop a wetting layer that is modestly rich in the B phase. Then the particle shell is the nanometric segregated layer of the A phase, and the core is the alloy that is modestly rich in the A phase.

Preconditioning and Iterative Solution of All-at-Once Systems for Evolutionary Partial Differential Equations

Standard Krylov subspace solvers for self-adjoint problems have rigorous convergence bounds based solely on eigenvalues. However, for non-self-adjoint problems, eigenvalues do not determine behavior even for widely used iterative methods. In this paper, we discuss time-dependent PDE problems, which are always non-self-adjoint. We propose a block circulant preconditioner for the all-at-once evolutionary PDE system which has block Toeplitz structure. Through reordering of variables to obtain a symmetric system, we are able to rigorously establish convergence bounds for MINRES which guarantee a number of iterations independent of the number of time-steps for the all-at-once system. If the spatial differential operators are simultaneously diagonalizable, we are able to quickly apply the preconditioner through use of a sine transform; and for those that are not, we are able to use an algebraic multigrid process to provide a good approximation. Results are presented for solution to both the heat and convection diffusion equations.

PDEModelica1: a Modelica language extension for partial differential equations implemented in OpenModelica

Modelica is a language for modeling complex physical systems. It supports algebraic and ordinary differential equations. This paper presents a Modelica language extension for one-dimensional evolutionary partial differential equations (PDE) called PDEModelica1, which is based on earlier extensions, establishes several enhancements, and its support is implemented in OpenModelica. PDEs are solved using the method of lines. In this paper, we discuss changes to and enhancements of previous extensions as well as the numerical methods used. To conclude, we present a counter-current heat exchange model with results to demonstrate the applicability of our extension.

Parametric excitation of an axisymmetric flow of a thin liquid film down a vertical fiber

We investigate axisymmetric flow of thin liquid films falling down the outer side of a vertical cylindrical surface oscillating harmonically in the axial direction. A set of two coupled nonlinear partial differential evolution equations in terms of the local instantaneous film thickness and volumetric flow rate is derived using the Galerkin method. A linear stability analysis of the time-periodic thickness-uniform base state is performed using the Floquet theory. We carry out numerical investigation of the nonlinear dynamics of the films in the framework of the set of evolution equations derived here. We find that the film evolution exhibits strange attractor dynamics ranging from coherent to fully irregular regimes. We show that harmonic parametric excitation affects the spatial topological structure of the interfacial wave and may modify its type from depression wave to elevation wave. The harmonic parametric excitation may also result in a significant decrease in the interfacial wave amplitude.

A new compact finite difference quasilinearization method for nonlinear evolution partial differential equations

Abstract This article presents a new method of solving partial differential equations. The method is an improvement of the previously reported compact finite difference quasilinearization method (CFDQLM) which is a combination of compact finite difference schemes and quasilinearization techniques. Previous applications of compact finite difference (FD) schemes when solving parabolic partial differential equations has been solely on discretizing the spatial variables and another numerical technique used to discretize temporal variables. In this work we attempt, for the first time, to use the compact FD schemes in both space and time. This ensures that the rich benefits of the compact FD schemes are carried over to the time variable as well, which improves the overall accuracy of the method. The proposed method is tested on four nonlinear evolution equations. The method produced highly accurate results which are portrayed in tables and graphs.

Spontaneous collective synchronization in the Kuramoto model with additional non-local interactions

In the context of the celebrated Kuramoto model of globally-coupled phase oscillators of distributed natural frequencies, which serves as a paradigm to investigate spontaneous collective synchronization in many-body interacting systems, we report on a very rich phase diagram in presence of thermal noise and an additional non-local interaction on a one-dimensional periodic lattice. Remarkably, the phase diagram involves both equilibrium and non-equilibrium phase transitions. In two contrasting limits of the dynamics, we obtain exact analytical results for the phase transitions. These two limits correspond to (i) the absence of thermal noise, when the dynamics reduces to that of a non-linear dynamical system, and (ii) the oscillators having the same natural frequency, when the dynamics becomes that of a statistical system in contact with a heat bath and relaxing to a statistical equilibrium state. In the former case, our exact analysis is based on the use of the so-called Ott-Antonsen ansatz to derive a reduced set of nonlinear partial differential equations for the macroscopic evolution of the system. Our results for the case of statistical equilibrium are on the other hand obtained by extending the well-known transfer matrix approach for nearest-neighbor Ising model to consider non-local interactions. The work offers a case study of exact analysis in many-body interacting systems. The results obtained underline the crucial role of additional non-local interactions in either destroying or enhancing the possibility of observing synchrony in mean-field systems exhibiting spontaneous synchronization.

Symmetric solutions of evolutionary partial differential equations

We show that for a large class of evolutionary nonlinear and nonlocal partial differential equations, symmetry of solutions implies very restrictive properties of the solutions and symmetry axes. These restrictions are formulated in terms of three principles, based on the structure of the equations. The first principle covers equations that allow for steady solutions and shows that any spatially symmetric solution is in fact steady with a speed determined by the motion of the axis of symmetry at the initial time. The second principle includes equations that admit breathers and steady waves, and therefore is less strong: it holds that the axes of symmetry are constant in time. The last principle is a mixed case, when the equation contains terms of the kind from both earlier principles, and there may be different outcomes; for a class of such equations one obtains that a spatially symmetric solution must be constant in both time and space. We list and give examples of more than 30 well-known equations and systems in one and several dimensions satisfying these principles; corresponding results for weak formulations of these equations may be attained using the same techniques. Our investigation is a generalisation of a local and one-dimensional version of the first principle from Ehrnström et al (2009 Int. Math. Res. Not. 2009 4578–96) to nonlocal equations, systems and higher dimensions, as well as a study of the standing and mixed cases.

Lie–Poisson structures over differential algebras

Based on key elements of Olver’s approach to partial differential equations for Hamiltonian evolution, we propose an algebraic construction appropriate for Hamiltonian evolutionary systems with constraints.

Numerical approximation using evolution PDE variational splines

This article deals with a numerical approximation method using an evolutionary partial differential equation (PDE) by discrete variational splines in a finite element space. To formulate the problem, we need an evolutionary PDE equation with respect to the time and the position, certain boundary conditions and a set of approximating points. We show the existence and uniqueness of the solution and we study a computational method to compute such a solution. Moreover, we established a convergence result with respect to the time and the position. We provided several numerical and graphic examples of approximation in order to show the validity and effectiveness of the presented method.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 34: 5–18, 2018

Partial differential variational inequalities involving nonlocal boundary conditions in Banach spaces

In this paper, we firstly introduce a complicated system obtained by mixing a nonlinear evolutionary partial differential equation and a mixed variational inequality in infinite dimensional Banach spaces in the case where the set of constraints is not necessarily bounded and the problem is driven by nonlocal boundary conditions, which is called partial differential variational inequality ((PDVI), for short). Then, we show that the solution set of the mixed variational inequality involved in problem (PDVI) is nonempty, bounded, closed and convex. Moreover, the upper semicontinuity and measurability properties for set-valued mapping U:[0,T]×E2→Cbv(E1) (see (3.7), below) are also established. Finally, several existence results for (PDVI) are obtained by using a fixed point theorem for condensing set-valued operators and theory of measure of noncompactness.

A Multi-Domain Bivariate Pseudospectral Method for Evolution Equations

In this paper, we present a new general approach for solving nonlinear evolution partial differential equations. The novelty of the approach is in the combination of spectral collocation and Lagrange interpolation polynomials with Legendre–Gauss–Lobatto grid points to descritize and solve equations in piece-wise defined intervals. The method is used to solve several nonlinear evolution partial differential equations, namely, the modified KdV–Burgers equation, modified KdV equation, Fisher’s equation, Burgers–Fisher equation, Burgers–Huxley equation and the Fitzhugh–Nagumo equation. The results are compared with known analytic solutions to confirm accuracy, convergence and to get a general understanding of the performance of the method. In all the numerical experiments, we report a high degree of accuracy of the numerical solutions. Strategies for implementing various boundary conditions are discussed.

Analysis of order reduction when integrating linear initial boundary value problems with Lawson methods

In this paper, a thorough analysis is given for the order which is observed when integrating evolutionary linear partial differential equations with Lawson methods. The analysis is performed under the general framework of C0-semigroups in Banach spaces and hence it can be applied to the numerical time integration of many initial boundary value problems which are described by linear partial differential equations. Conditions of regularity and annihilation at the boundary of these problems are then stated to justify the precise order which is observed, including fractional order of convergence.

Generalised Dirichlet to Neumann maps for linear dispersive equations on half-line

AbstractA large class of initial-boundary problems of linear evolution partial differential equations formulated on the half-line is analyzed via the unified transform method. In particular, explicit representations are presented for the generalised Dirichlet to Neumann maps. Namely, the determination of the unknown boundary values when an essential set of initial and boundary data is given.