Methods For Differential Equations Research Articles

Superconvergence and Postprocessing of Collocation Methods for Fractional Differential Equations

Superconvergence and Postprocessing of Collocation Methods for Fractional Differential Equations

Convergence and stability of an explicit method for nonlinear stochastic differential equations with piecewise continuous arguments

This paper constructs a new explicit method for stochastic differential equations with piecewise continuous arguments (SDEPCAs), where the drift coefficients grow superlinearly and the diffusion coefficients have at most linear growth. We show that this method converges strongly with the convergence rate 1/2 to the exact solutions of SDEPCAs over the finite time interval and demonstrate it can inherit the mean square exponential stability of the underlying SDEPCAs. Several numerical experiments are carried out to support our findings.

A new recovery based [formula omitted] element method for fourth-order equations

In this paper, we study a new Hessian recovery based method for fourth-order differential equations by combining finite element methods (FEMs) and finite difference methods (FDMs). The new C0 finite element method is established under the framework of Petrov–Galerkin methods. Meanwhile, some unsuccessful higher-order methods based on the Hessian recovery operator are also discussed. Finally, we extend our method to solve Cahn-Hilliard equations. Numerical results validate that our proposed C0 method works well for Cahn-Hilliard, with good convergence property and easy implementation.

Efficient Iterative Arbitrary High-Order Methods: an Adaptive Bridge Between Low and High Order

AbstractWe propose a new paradigm for designing efficient p-adaptive arbitrary high-order methods. We consider arbitrary high-order iterative schemes that gain one order of accuracy at each iteration and we modify them to match the accuracy achieved in a specific iteration with the discretization accuracy of the same iteration. Apart from the computational advantage, the newly modified methods allow to naturally perform the p-adaptivity, stopping the iterations when appropriate conditions are met. Moreover, the modification is very easy to be included in an existing implementation of an arbitrary high-order iterative scheme and it does not ruin the possibility of parallelization, if this was achievable by the original method. An application to the Arbitrary DERivative (ADER) method for hyperbolic Partial Differential Equations (PDEs) is presented here. We explain how such a framework can be interpreted as an arbitrary high-order iterative scheme, by recasting it as a Deferred Correction (DeC) method, and how to easily modify it to obtain a more efficient formulation, in which a local a posteriori limiter can be naturally integrated leading to the p-adaptivity and structure-preserving properties. Finally, the novel approach is extensively tested against classical benchmarks for compressible gas dynamics to show the robustness and the computational efficiency.

Functional‐preserving predictor‐corrector multiderivative schemes

AbstractIn this work, we develop a class of high‐order multiderivative time integration methods that is able to preserve certain functionals discretely. Important ingredients are the recently developed Hermite‐Birkhoff‐Predictor‐Corrector (HBPC) methods and the technique of relaxation for numerical methods of ordinary differential equations (ODEs). We explain the algorithm in detail and show numerical results for two‐ and three‐derivative methods, comparing relaxed and unrelaxed methods. The numerical results demonstrate that, at the slight cost of the relaxation, an improved scheme is obtained.

CONVERGENCE AND STABILITY OF GALERKIN FINITE ELEMENT METHOD FOR HYPERBOLIC PARTIAL DIFFERENTIAL EQUATION WITH PIECEWISE CONTINUOUS ARGUMENTS OF ADVANCED TYPE

This paper deals with the convergence and stability of Galerkin finite element method for a hyperbolic partial differential equations with piecewise continuous arguments of advanced type. First of all, we obtain the expression of analytic solution by the method of separation variable, then the sufficient conditions for stability are obtained. Semidiscrete and fully discrete schemes are derived by Galerkin finite element method, and their convergence are both analyzed in L2-norm. Moreover, the stability of the two schemes are investigated. The semidiscrete scheme can achieve unconditionally stability. The sufficient conditions of stability for fully discrete scheme are derived under which the analytic solution is asymptotically stable. Finally, some numerical experiments are presented to illustrate the theoretical results.

Data-driven simulation of ammonia combustion using neural ordinary differential equations (NODE)

The direct use of detailed chemical kinetics in combustion simulations is limited by the extremely high computational costs. Recently, Owoyele and Pal (Energy and AI, 2022), proposed the neural ordinary differential equations (NODE) method to accelerate calculations of chemical kinetics and proved its effectiveness in zero-dimensional calculations of hydrogen combustion considering 9 species and 21 reactions. However, its performance for more realistic high-dimensional calculations and more complex kinetic systems remains unexplored. Therefore, this study further applies the method for more complex chemical kinetics of ammonia combustion, especially with optimizations in the data sampling, model training strategies, and model application methods that remedy the problems of versatility and application to more practical simulations. The newly developed NODE models are comprehensively validated in the zero-dimensional calculations of ammonia auto-ignition, one-dimensional calculations of laminar freely-propagating ammonia-premixed flames, and two-dimensional direct numerical simulation (DNS) of ammonia-premixed flames in a temporally evolving jet. Present NODE models focus on seven chemical species, namely NH3, O2, H2, OH, H2O, N2, and NO, and the results show that, compared with the results obtained by using detailed chemical kinetics, this method is able to reduce the computational costs of the zero-dimensional auto-ignition reaction to 1/24 while reproducing the ignition delay time for a wide range of initial temperatures and equivalence ratios with relatively good accuracy. Additionally, the method is able to reduce the computational costs of the one-dimensional freely propagating flame and two-dimensional jet flame to 1/4 and 1/38 respectively, while acceptable reproduction of the laminar flame speed and temporal evolution of the gas temperature and mass fractions of the interested species can be achieved.

An accurate and robust Eulerian finite element method for partial differential equations on evolving surfaces

An accurate and robust Eulerian finite element method for partial differential equations on evolving surfaces

Plants and age-structured pest dynamics with natural enemy as control strategy: A fractional differential equations model

In recent years, plant pests have drawn much attention from researchers. Biological control proved an efficient replacement for chemical and other pest control techniques. The present study proposes the cognition of an immature pest population unaffected by the natural enemy and grows exponentially by the harvesting effort of the mature pest population to plants. The fractional calculus environment proposes A four-compartment: plant, immature and mature pest, and natural enemy model. Equilibrium points and the system’s stability are studied theoretically and numerically. Simulation study for different values of fractional order generates the optimum solution. The suggested method is much more benignant than the existing method of ordinary differential equations as we can figure out the infrequent changes in the variable using a fractional derivative. Creating the memory effect in the system using fractional derivatives makes it more realistic.

Face sheet defects in plastic forming of aluminum foam sandwich panel: theoretical models, numerical and experimental studies

In this paper, the theoretical models of face wrinkling and dimpling defects occurred in the plastic forming of aluminum foam sandwich panel (AFSP) were constructed and the influences of AFSP parameters on defect were investigated to predict and avoid the surface forming defects. The wrinkling and dimpling behaviors of shaped AFSPs were analyzed at different scales first, and the analytical models of critical wrinkling and dimpling stresses of face sheet for the forming of AFSP were developed using energy method and differential equilibrium equation method respectively. Finite element simulations of curved AFSPs forming were performed using macroscopic and mesoscopic AFSP models, and plastic forming experiments of spherical and saddle-shaped AFSPs with different core relative densities were conducted. The surface profiles of simulated parts and the face wrinkles and dimples in experimental parts indicated that the established theoretical models can predict the occurrence of face sheet defects in curved AFSPs forming accurately. Furthermore, the effects of geometric and material parameters of AFSP on face wrinkling suggested that the seriousness of face wrinkling can be reduced by decreasing the thicknesses of face sheet and core, and the wrinkling can be suppressed when the elastic Poisson's ratio of core is close to 0 and 0.5.

Collocation method for stochastic delay differential equations

In this work, we present a collocation-based numerical approach for handling stochastic delay differential equations. We approximate the solution function, and after that, we carry out integrations between the predefined collocation points to achieve a mapping from the delayed state to the present state. We build the first and second moment mapping matrices based on the mapping, and we utilize the matrices to approximate the stationary first and second moments and their stability. Numerical studies of a first and second-order stochastic delay differential equation show the convergence and time complexity of the stochastic collocation method. The last section covers the issues and possible further improvements of the method.

A novel analytical method for the multi-delay fractional differential equations through the matrix of clique polynomials of the cocktail party graph

This paper provides a unique graph matrix approach based on the Clique polynomials of the Cocktail party graph to effectively solve multi-delay fractional differential equations (MDFDE) with variable coefficients. Clique polynomials of the cocktail party graph’s basis are used as a trial function for the solution of MDFDE. An operational matrix of Caputo fractional derivative of polynomial approximation is employed to transform the MDFDEs into a system of algebraic equations. The unknowns in the system are determined by Newton’s method in Mathematica. The productivity of the suggested strategy is examined with five numerical examples. From all the discussed examples, it is observed that the proposed Clique polynomial collocation method (CPCM) has excellent compatibility with linear MDFDEs by attaining an exact solution. To substantiate the validity of the recommended method, obtained results with comparisons, computation time, convergence analysis, error estimations, and graphical illustrations are offered.

Metalearning Generalizable Dynamics from Trajectories.

We present the interpretable meta neural ordinary differential equation (iMODE) method to rapidly learn generalizable (i.e., not parameter-specific) dynamics from trajectories of multiple dynamical systems that vary in their physical parameters. The iMODE method learns metaknowledge, the functional variations of the force field of dynamical system instances without knowing the physical parameters, by adopting a bilevel optimization framework: an outer level capturing the common force field form among studied dynamical system instances and an inner level adapting to individual system instances. Apriori physical knowledge can be conveniently embedded in the neural network architecture as inductive bias, such as conservative force field and Euclidean symmetry. With the learned metaknowledge, iMODE can model an unseen system within seconds, and inversely reveal knowledge on the physical parameters of a system, or as a neural gauge to "measure" the physical parameters of an unseen system with observed trajectories. iMODE can be generally applied to a dynamical system of an arbitrary type or number of physical parameters and is validated on bistable, double pendulum, Van der Pol, Slinky, and reaction-diffusion systems.

Chaplygin’s method for second-order neutral differential equations with piecewise constant deviating arguments

Chaplygin’s method for second-order neutral differential equations with piecewise constant deviating arguments

The influence of fracture surface morphology on nonuniform etching in limestone acid fracturing

The success of acid fracturing in tight limestone reservoirs relies on the formation of nonuniform etching on the fracture wall. However, almost all acid fracturing models neglect the effect of fracture surface morphology on nonuniform etching. In this study, a three-dimensional acid fracturing mathematical model is developed to quantify and analyze the influence of rough fracture surface morphology on nonuniform etching for the first time by defining a morphology coefficient to describe the degree of local and overall morphological undulations of the fracture surface from a three-dimensional scale. The model uses the moving mesh partial differential equation (MMPDE) method to describe the flow field and spatial variation in the fracture as the fracture surface is continuously dissolved with the acid during acid fracturing, and the MMPDE moving mesh is coupled with the flow field, mass transfer, acid-rock reaction and fracture width variation. Based on the model, the effects of flow pattern, morphology coefficient, initial fracture width and fracture type on the nonuniform etching are studied. The simulation results demonstrate that the rough fracture surface can disturb the flow field and make the acid distribution more turbulent, which is conducive to the formation of nonuniform etching on the fracture surface. This phenomenon of flow rate differentiation can enhance the acid transfer distance. The local etching degree of the fracture surface decreases with the increase of the morphology coefficient of the area, and the strongly etching areas will appear around it. When the morphology coefficient of the local fracture area is larger than 1.2218, the area can appear obvious strongly etching areas and weakly etching areas after acid etching. The larger the initial fracture width is, the smaller the degree of nonuniform etching on the fracture surface. The prediction model constructed based on multiple linear regression was highly accurate (R2 = 0.9268) for the standard deviation of facture width (StdWidth) after acid etching. The critical value of StdWidth is recommended to be 0.6 as the criterion for judging the nonuniform etching ability. When the nonuniformity of the fracture surface is greater than this value, it is considered that the nonuniform etching ability is strong. In addition, the fracture with shear damage can produce relative sliding distance, which makes the fracture morphology on both sides of the fracture surface unevenly distributed and thus disturbs the flow field, which enhances the degree of nonuniform etching on the fracture surface. As the sliding distance increase, the magnitude of this enhancement also increases gradually. This study proposes a new three-dimensional characterization method for rough fracture surface morphology, solves the issue of mismatching flow field due to dynamic change in fracture width during acid fracturing.

Editorial for the Special Issue “Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms”

This editorial text is a short introductory guide to the book edition of the Special Issue “Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms”, which was published in the MDPI journal Mathematics in the years 2022–2023 [...]

Solution method and parameter estimation of uncertain partial differential equation with application to China’s population

Solution method and parameter estimation of uncertain partial differential equation with application to China’s population

Computational Multiscale Methods for Nondivergence-Form Elliptic Partial Differential Equations

Abstract This paper proposes novel computational multiscale methods for linear second-order elliptic partial differential equations in nondivergence form with heterogeneous coefficients satisfying a Cordes condition. The construction follows the methodology of localized orthogonal decomposition (LOD) and provides operator-adapted coarse spaces by solving localized cell problems on a fine scale in the spirit of numerical homogenization. The degrees of freedom of the coarse spaces are related to nonconforming and mixed finite element methods for homogeneous problems. The rigorous error analysis of one exemplary approach shows that the favorable properties of the LOD methodology known from divergence-form PDEs, i.e., its applicability and accuracy beyond scale separation and periodicity, remain valid for problems in nondivergence form.

Asymptotic mean square stability of Predictor-Corrector methods for stochastic delay ordinary and partial differential equations

In this paper, we focus on the asymptotic mean square stability of stochastic delay ordinary and partial differential equations. By virtue of root locus technique, the sufficient and necessary conditions of asymptotic mean square stability for Predictor-Corrector methods are obtained, which are suitable for solving above two types of problems. Furthermore, by regulating the value of parameter θ in drift term of the schemes, we could investigate a series of schemes to explore different size of stable regions, this amounts to provide a road to obtain the optimal stable regions. Several theorems prove that the Predictor-Corrector schemes could inherit the asymptotic mean square stability for above two original problems. Numerical experiments verify the stability theorems, precision and the parallelism.

Multi-dimensional summation-by-parts operators for general function spaces: Theory and construction

Summation-by-parts (SBP) operators allow us to systematically develop energy-stable and high-order accurate numerical methods for time-dependent differential equations. Until recently, the main idea behind existing SBP operators was that polynomials can accurately approximate the solution, and SBP operators should thus be exact for them. However, polynomials do not provide the best approximation for some problems, with other approximation spaces being more appropriate. We recently addressed this issue and developed a theory for one-dimensional SBP operators based on general function spaces, coined function-space SBP (FSBP) operators. In this paper, we extend the theory of FSBP operators to multiple dimensions. We focus on their existence, connection to quadratures, construction, and mimetic properties. A more exhaustive numerical demonstration of multi-dimensional FSBP (MFSBP) operators and their application will be provided in future works. Similar to the one-dimensional case, we demonstrate that most of the established results for polynomial-based multi-dimensional SBP (MSBP) operators carry over to the more general class of MFSBP operators. Our findings imply that the concept of SBP operators can be applied to a significantly larger class of methods than is currently done. This can increase the accuracy of the numerical solutions and/or provide stability to the methods.