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The current article is devoted to proposing a local meshfree method for simulating the unsteady diffusion–convection problems of anisotropic functionally graded materials (AFGM). The first stage of the proposed numerical algorithm is based on discretizing the spatial variables by utilizing the direct meshless local Petrov–Galerkin (DMLPG) method. Then a system of ODEs is extracted which is associated to the time variable. Furthermore, the fourth-order exponential time differencing Runge–Kutta method is applied to solve the system of ODEs. The main aim of this work is to introduce a flexible and low-cost numerical procedure to solve unsteady diffusion–convection problems of AFGM on the complicated geometry. The new numerical technique has admissible resultants to simulate the studied model. This numerical formulation is applied to several examples with complex domains to check its ability. [ABSTRACT FROM AUTHOR]

The current study examines the (2 + 1)-dimensional fractional Chaffee–Infante (FCI) model, which is a nonlinear evolution equation that characterizes the processes of pattern generation, reaction-diffusion, and nonlinear wave propagation. The construction of analytical solutions involves the use of analytical methods, namely the Khater III and improved Kudryashov schemes. The He’s Variational Iteration method is employed as a numerical approach to validate the accuracy of the obtained solutions. The main objective of this study is to get novel analytical and numerical solutions for the FCI model, with the intention of gaining a deeper understanding of the system’s dynamics and its possible implications in the fields of fluid mechanics, plasma physics, and optical fiber communications. The study makes a valuable contribution to the area of nonlinear science via the use of innovative analytical and numerical methodologies in the FCI model. This research enhances our comprehension of pattern creation, reaction–diffusion phenomena, and the propagation of nonlinear waves in diverse physical scenarios. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new method for computing the stray-field and the corresponding energy for a given magnetization configuration. Our approach is based on the use of inverted finite elements and does not need any truncation. After analyzing the problem in an appropriate functional framework, we describe the method and we prove its convergence. We then display some computational results which demonstrate its efficiency and confirm its full potential. [ABSTRACT FROM AUTHOR]

POD-DL-ROMs have been recently proposed as an extremely versatile strategy to build accurate and reliable reduced order models (ROMs) for nonlinear parametrized partial differential equations, combining (i) a preliminary dimensionality reduction obtained through proper orthogonal decomposition (POD) for the sake of efficiency, (ii) an autoencoder architecture that further reduces the dimensionality of the POD space to a handful of latent coordinates, and (iii) a dense neural network to learn the map that describes the dynamics of the latent coordinates as a function of the input parameters and the time variable. Within this work, we aim at justifying the outstanding approximation capabilities of POD-DL-ROMs by means of a thorough error analysis, showing how the sampling required to generate training data, the dimension of the POD space, and the complexity of the underlying neural networks, impact on the solutions us to formulate practical criteria to control the relative error in the approximation of the solution field of interest, and derive general error estimates. Furthermore, we show that, from a theoretical point of view, POD-DL-ROMs outperform several deep learning-based techniques in terms of model complexity. Finally, we validate our findings by means of suitable numerical experiments, ranging from parameter-dependent operators analytically defined to several parametrized PDEs. [ABSTRACT FROM AUTHOR]

In this paper, we propose a reduced-order modeling strategy for two-way Dirichlet–Neumann parametric coupled problems solved with domain-decomposition (DD) sub-structuring methods. We split the original coupled differential problem into two sub-problems with Dirichlet and Neumann interface conditions, respectively. After discretization by, e.g., the finite element method, the full-order model (FOM) is solved by Dirichlet–Neumann iterations between the two sub-problems until interface convergence is reached. We then apply the reduced basis (RB) method to obtain a low-dimensional representation of the solution of each sub-problem. Furthermore, we apply the discrete empirical interpolation method (DEIM) at the interface level to achieve a fully reduced-order representation of the DD techniques implemented. To deal with non-conforming FE interface discretizations, we employ the INTERNODES method combined with the interface DEIM reduction. The reduced-order model (ROM) is then solved by sub-iterating between the two reduced-order sub-problems until the convergence of the approximated high-fidelity interface solutions. The ROM scheme is numerically verified on both steady and unsteady coupled problems, in the case of non-conforming FE interfaces. [ABSTRACT FROM AUTHOR]

Recently, physics-informed neural networks (PINNs) have offered a powerful new paradigm for solving problems relating to differential equations. Compared to classical numerical methods, PINNs have several advantages, for example their ability to provide mesh-free solutions of differential equations and their ability to carry out forward and inverse modelling within the same optimisation problem. Whilst promising, a key limitation to date is that PINNs have struggled to accurately and efficiently solve problems with large domains and/or multi-scale solutions, which is crucial for their real-world application. Multiple significant and related factors contribute to this issue, including the increasing complexity of the underlying PINN optimisation problem as the problem size grows and the spectral bias of neural networks. In this work, we propose a new, scalable approach for solving large problems relating to differential equations called finite basis physics-informed neural networks (FBPINNs). FBPINNs are inspired by classical finite element methods, where the solution of the differential equation is expressed as the sum of a finite set of basis functions with compact support. In FBPINNs, neural networks are used to learn these basis functions, which are defined over small, overlapping subdomains. FBINNs are designed to address the spectral bias of neural networks by using separate input normalisation over each subdomain and reduce the complexity of the underlying optimisation problem by using many smaller neural networks in a parallel divide-and-conquer approach. Our numerical experiments show that FBPINNs are effective in solving both small and larger, multi-scale problems, outperforming standard PINNs in both accuracy and computational resources required, potentially paving the way to the application of PINNs on large, real-world problems. [ABSTRACT FROM AUTHOR]

Extreme learning machine (ELM) is a type of randomized neural networks originally developed for linear classification and regression problems in the mid-2000s, and has recently been extended to computational partial differential equations (PDE). This method can yield highly accurate solutions to linear/nonlinear PDEs, but requires the last hidden layer of the neural network to be wide to achieve a high accuracy. If the last hidden layer is narrow, the accuracy of the existing ELM method will be poor, irrespective of the rest of the network configuration. In this paper we present a modified ELM method, termed HLConcELM (hidden-layer concatenated ELM), to overcome the above drawback of the conventional ELM method. The HLConcELM method can produce highly accurate solutions to linear/nonlinear PDEs when the last hidden layer of the network is narrow and when it is wide. The new method is based on a type of modified feedforward neural networks (FNN), termed HLConcFNN (hidden-layer concatenated FNN), which incorporates a logical concatenation of the hidden layers in the network and exposes all the hidden nodes to the output-layer nodes. HLConcFNNs have the interesting property that, given a network architecture, when additional hidden layers are appended to the network or when extra nodes are added to the existing hidden layers, the representation capacity of the HLConcFNN associated with the new architecture is guaranteed to be not smaller than that of the original network architecture. Here representation capacity refers to the set of all functions that can be exactly represented by the neural network of a given architecture. We present ample benchmark tests with linear/nonlinear PDEs to demonstrate the computational accuracy and performance of the HLConcELM method and the superiority of this method to the conventional ELM from previous works. [ABSTRACT FROM AUTHOR]

In this manuscript, a numerical method based on the conjunction of Paraskevopoulos's algorithm and operational matrices is developed to solve numerically the multi-order linear and nonlinear fractional differential equations. By means of this conjunction, the multi-order problem is decomposed into a system of differential equations of fractional order which are then solved by employing operational matrices approach. The accuracy and efficiency of the method is examined by taking some examples. In addition, the numerical results presented in Pak et al. [Analytical solutions of linear inhomogeneous fractional differential equation with continuous variable coefficients. Adv Differ Equ. 2019;2019(256):1–22] are improved in our work.

In this study, high accurate numerical solutions of the modified Korteweg-de Vries equation via contribution of two effective methods and efficient linearizing technique are obtained. In order to observe the performance of the present approximation, five famous applications will be handled. To see the position of the new method in the scientific literature, all numerical results obtained by using the same parameters with earlier applications and compared with each other. Furthermore, to display the numerical behaviour of the shallow water waves, all of the applications are illustrated in detail and given in 2D and 3D. Numerical results show that present algorithm will be useful for the other nonlinear scientific problems. [ABSTRACT FROM AUTHOR]

In this paper, we study a high-order unconditionally stable numerical method to approximate the class of multi-term time-fractional diffusion equations. This type of problem appears in the modelling of transport of certain quantities such as heat, mass, energy, solutes in ground water and soils. The multi-term time-fractional derivative is approximated by using the Crank–Nicolson method for the Caputo's time derivative. The space derivative is approximated by using the collocation method based on quintic B-spline basis functions. We have established the stability and convergence analysis of the proposed numerical scheme thoroughly, and it is shown that the order of convergence in space variable is almost four and in the time variable is O (Δ t 2 − max { γ , γ i }). To prove the accuracy and efficiency of the developed method, we consider four numerical examples and perform the numerical simulation. The developed algorithm works well andvalidate the theoretical results. The developed method is fourth-order convergent in the space variable, which is almost two orders of magnitude higher than the other spline collocation methods. [ABSTRACT FROM AUTHOR]

In this paper, some linearly implicit modified energy-conserving schemes are proposed for the modified Korteweg-de Vries equation (mKdV). The proposed schemes are based on the recently developed invariant energy quadratization (IEQ) approach and the scalar auxiliary variable (SAV) approach. We first introduce an auxiliary variable to transform the original model into an equivalent system, with a modified energy functional law. Then, the Fourier pseudospectral method is employed for the spatial discretization, and Crank-Nicolson, and Leap-Frog methods are used for the temporal discretization. We analyze the conservation properties, existence and uniqueness and the linear stability of the proposed schemes. The optimal order convergence rate of the semi-discrete scheme and the fully discrete schemes were analyzed, respectively. At last, some numerical examples are presented to illustrate the effectiveness of the proposed schemes. [ABSTRACT FROM AUTHOR]

Physics-informed neural networks approximate solutions of PDEs by minimizing pointwise residuals. We derive rigorous bounds on the error, incurred by PINNs in approximating the solutions of a large class of linear parabolic PDEs, namely Kolmogorov equations that include the heat equation and Black-Scholes equation of option pricing, as examples. We construct neural networks, whose PINN residual (generalization error) can be made as small as desired. We also prove that the total L2-error can be bounded by the generalization error, which in turn is bounded in terms of the training error, provided that a sufficient number of randomly chosen training (collocation) points is used. Moreover, we prove that the size of the PINNs and the number of training samples only grow polynomially with the underlying dimension, enabling PINNs to overcome the curse of dimensionality in this context. These results enable us to provide a comprehensive error analysis for PINNs in approximating Kolmogorov PDEs. [ABSTRACT FROM AUTHOR]

Our main concern in this study is to present the generalized results to investigate the existence of solutions to nonlinear fractional boundary value problems (FBVPs) with generalized nonlinear boundary conditions. The framework of the presented results relies on the lower and upper solutions approach which allows us to ensure the existence of solutions in a sector defined by well-ordered coupled lower and upper solutions. It is worth mentioning that the presented results unify the existence criteria of certain problems which were treated on a case-by-case basis in the literature. Two examples are supplied to support the results. [ABSTRACT FROM AUTHOR]

Summation-By-Parts (SBP) methods provide a systematic way of constructing provably stable numerical schemes. However, many proofs of convergence and accuracy rely on the assumption that the SBP operator possesses a particular eigenvalue property. In this note, three results pertaining to this property are proven. Firstly, the eigenvalue property does not hold for all nullspace consistent SBP operators. Secondly, this issue can be addressed without affecting the accuracy of the method by adding a specially designed, arbitrarily small perturbation term to the SBP operator. Thirdly, all pseudospectral methods satisfy the eigenvalue property. [ABSTRACT FROM AUTHOR]

In this paper, we study and construct a higher order numerical algorithm for singularly perturbed differential-difference convection–diffusion equation with retarded term appearing in computational biological science. The solution of the considered class of problems may exhibit a boundary layer due to the presence of the perturbation parameter and retarded term. We have discussed the analytical behaviour of the exact solution and its partial derivatives. We have used Taylor's series expansion to approximate the advance and delay terms of the model problem and then the problem is discretized using Crank–Nicolson's method on equidistant mesh in the time direction and modified cubic B-spline basis functions on generalized Shishkin mesh in the spatial direction. We have also shown that the developed algorithm is unconditionally stable. The proposed numerical algorithm is proved to be ε -uniformly convergent of order four in the spatial direction up to a logarithmic factor and second-order convergent in the time direction. To demonstrate the accuracy and to validate the theoretical results of the proposed numerical algorithm, we have presented three numerical experiments. We have also compared our method with existing schemes in the literature to prove the accuracy of the proposed numerical algorithm. [ABSTRACT FROM AUTHOR]

Differential equation (DE) is a commonly used modeling method in various scientific subjects such as finance and biology. The parameters in DE models often have interesting scientific interpretations, but their values are often unknown and need to be estimated from the measurements. Here, we develop a one-stage parameter estimation framework, which is based on the Markov Chain Monte Carlo (MCMC) method, to draw the samples from the posterior distribution of the unknown parameters from the given noisy and scarce observations of the solution only. A likelihood function including a novel potential is constructed to infer the unknown parameters and the novel potential works by measuring the residual errors of both data and DE model with given model parameters. A key issue in parameter estimation problem is to robustly estimate the solution and its derivatives from noisy observations of only the function values at given location points, under the assumption of a physical model in the form of differential equation governing the function and its derivatives. To address the key issue, we propose to use the Gaussian process regression with constraint (GPRC) method which jointly model the solution, its derivatives, and the parametric differential equation, to estimate the solution and its derivatives. With numerical examples, we illustrate that the proposed method has competitive performance against existing approaches for estimating the unknown parameters in DEs. [ABSTRACT FROM AUTHOR]

In this paper, we present a nonconforming immersed finite element method for solving elliptic optimal control problems with interfaces. The immersed finite element space is constructed based on the rotated-Q1 nonconforming finite elements with the integral-value degrees of freedom. The method can overcome the difficulties encountered by conforming immersed finite element method. We derive error estimates for the control, state and adjoint state in both energy and L 2 norms. Numerical results are reported to support the theoretical analysis. [ABSTRACT FROM AUTHOR]

One way to deliver a drug and control its release is to use magnetic nanoparticles as drug carriers and conduct them by a magnetic field. In this method, drugs and magnetic nanoparticles (usually based on iron, nickel, cobalt or their oxides) are designed and prepared in a combination and injected into the bloodstream upstream of the tissue. These nanoparticles move through the bloodstream and approach the magnetic field. In this area, a magnetic field is applied to these nanoparticles in order to be absorbed and cause the accumulation of these carriers and increase their concentration in the vessel wall. A coupled non-Fickian model for targeted drug delivery from vessel wall into the tumor tissue using magnetic nanoparticle is proposed. The model takes into account all effective fields (chemical, mechanical and magnetic) and other contributed fields such as convection due to bloodstream. In this work, the effect of viscoelasticity on the nanoparticles concentration release in the tumor in the presence of a magnetic field is studied. The influence of the parameters on the transport of the nanoparticles concentration from the vessel wall into the tumor tissue is analysed. The stability analysis of the model under the initial perturbations is studied. [ABSTRACT FROM AUTHOR]

In the context of coupling hyperbolic problems, the maximum stable time step of an explicit numerical scheme may depend on the design of the coupling procedure. If this is the case, the coupling procedure is sensitive to changes in model parameters independent of the Courant–Friedrichs–Levy condition. This sensitivity can cause artificial stiffness that degrades the performance of a numerical scheme. To overcome this problem, we present a systematic and general procedure for weakly imposing coupling conditions via penalty terms in a provably non-stiff manner. The procedure can be used to construct both energy conservative and dissipative couplings, and the user is given control over the amount of dissipation desired. The resulting formulation is simple to implement and dual consistent. The penalty coefficients take the form of projection matrices based on the coupling conditions. Numerical experiments demonstrate that this procedure results in both optimal spectral radii and superconvergent linear functionals. [ABSTRACT FROM AUTHOR]

In this manuscript, a numerical method based on the conjunction of Paraskevopoulos's algorithm and operational matrices is developed to solve numerically the multi-order linear and nonlinear fractional differential equations. By means of this conjunction, the multi-order problem is decomposed into a system of differential equations of fractional order which are then solved by employing operational matrices approach. The accuracy and efficiency of the method is examined by taking some examples. In addition, the numerical results presented in Pak et al. [Analytical solutions of linear inhomogeneous fractional differential equation with continuous variable coefficients. Adv Differ Equ. 2019;2019(256):1–22] are improved in our work. [ABSTRACT FROM AUTHOR]

We study a liquid droplet moving inside a rarefied gas. In other words, we consider a two-phase flow with liquid and rarefied gas phases and an interface between the two phases which deforms with respect to time and space. The gas phase is modelled by the Bhatanager–Gross–Krook (BGK) model of the Boltzmann equation. The liquid phase is modelled by the incompressible Navier–Stokes equations. Interface boundary conditions for the liquid and gas phases are presented. The BGK model is solved by a semi-Lagrangian scheme with a meshfree reconstruction procedure. A similar meshfree particle method is used to solve the incompressible Navier–Stokes equations for the liquid phase. In 1D the solutions of the BGK-Navier-Stokes equations and the Boltzmann-Navier-Stokes equations are compared, where the Boltzmann equation is solved by a direct simulation Monte Carlo method. Results in 1 D and 2 D physical spaces are presented. [ABSTRACT FROM AUTHOR]

We analyze the problem of reconstruction of a bandlimited function f from the space–time samples of its states f t = ϕ t ∗ f resulting from the convolution with a kernel ϕ t . It is well-known that, in natural phenomena, uniform space–time samples of f are not sufficient to reconstruct f in a stable way. To enable stable reconstruction, a space–time sampling with periodic nonuniformly spaced samples must be used as was shown by Lu and Vetterli. We show that the stability of reconstruction, as measured by a condition number, controls the maximal gap between the spacial samples. We provide a quantitative statement of this result. In addition, instead of irregular space–time samples, we show that uniform dynamical samples at sub-Nyquist spatial rate allow one to stably reconstruct the function f ^ away from certain, explicitly described blind spots. We also consider several classes of finite dimensional subsets of bandlimited functions in which the stable reconstruction is possible, even inside the blind spots. We obtain quantitative estimates for it using Remez-Turán type inequalities. En route, we obtain Remez-Turán inequality for prolate spheroidal wave functions. To illustrate our results, we present some numerics and explicit estimates for the heat flow problem. [ABSTRACT FROM AUTHOR]

This study provides a new method for approximating the derivatives by cubic multiquadric quasi-interpolation operator. We prove that cubic multiquadric quasi-interpolation can approximate the derivative f (k) (x) with a high-order accuracy. Then, the stability of the method is analyzed by compared with the classical divided difference. Some numerical examples are chosen to demonstrate the accuracy and effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we study an efficient and high-order numerical technique based on the quartic trigonometric B-spline collocation method with quasilinearization approach to compute approximate solution of Lane–Emden equations subjected to Neumann and Robin boundary conditions arising in various physical models. The error analysis and convergence properties of the method are studied via Green's function approach, which shows that proposed method converges very fast. The advantage of this technique is its robustness and powerful experimental applicability for various types of boundary conditions. We provided several examples, to substantiate its rate of convergence. The numerical results are compared with exact solutions and some existing methods which demonstrate that our method gives more accurate results. Through the figure, we confirm the productivity and applicability of the method. [ABSTRACT FROM AUTHOR]

Optimal continuous-discrete filtering for a nonlinear system requires evolving the forward Kolmogorov equation, that is a Fokker–Planck equation, in alternation with Bayes' conditional updating. We present two numerical grid-methods that represent density functions on a mesh, or grid. For low-dimensional, smooth systems the finite-volume method is an effective solver that gives estimates that converge to the optimal continuous-time values. We give numerical examples to show that this finite-volume filter is able to handle multi-modal filtering distributions that result from rank-deficient observations, and that Bayes-optimal parameter estimation may be performed within the filtering process. The naïve discretization of density functions used in the finite-volume filter leads to an exponential increase of computational cost and storage with increasing dimension, that makes the finite-volume filter unfeasible for higher-dimensional problems. We circumvent this 'curse of dimensionality' by using a tensor train representation (or approximation) of density functions and employ an efficient implicit PDE solver that operates on the tensor train representation. We present numerical examples of tracking n weakly coupled pendulums in continuous time to demonstrate filtering with complex density functions in up to 80 dimensions. [ABSTRACT FROM AUTHOR]