Abstract: The purpose of this study is to introduce the reproducing kernel algorithm for treating classes of time-fractional partial differential equations subject to Robin boundary conditions with parameters derivative arising in fluid flows, fluid dynamics, groundwater hydrology, conservation of energy, heat conduction and electric circuit.,The method provides appropriate representation of the solutions in convergent series formula with accurately computable components. This representation is given in the W(Ω) and H(Ω) inner product spaces, while the computation of the required grid points relies on the R(y,s) (x, t) and r(y,s) (x, t) reproducing kernel functions.,Numerical simulation with different order derivatives degree is done including linear and nonlinear terms that are acquired by interrupting the n-term of the exact solutions. Computational results showed that the proposed algorithm is competitive in terms of the quality of the solutions found and is very valid for solving such time-fractional models.,Future work includes the application of the reproducing kernel algorithm to highly nonlinear time-fractional partial differential equations such as those arising in single and multiphase flows. The results will be published in forthcoming papers.,The study included a description of fundamental reproducing kernel algorithm and the concepts of convergence, and error behavior for the reproducing kernel algorithm solvers. Results obtained by the proposed algorithm are found to outperform in terms of accuracy, generality and applicability.,Developing analytical and numerical methods for the solutions of time-fractional partial differential equations is a very important task owing to their practical interest.,This study, for the first time, presents reproducing kernel algorithm for obtaining the numerical solutions of some certain classes of Robin time-fractional partial differential equations. An efficient construction is provided to obtain the numerical solutions for the equations, along with an existence proof of the exact solutions based upon the reproducing kernel theory.

Abstract: We present three schemes for the numerical approximation of fractional diffusion, which build on different definitions of such a non-local process. The first method is a PDE approach that applies to the spectral definition and exploits the extension to one higher dimension. The second method is the integral formulation and deals with singular non-integrable kernels. The third method is a discretization of the Dunford–Taylor formula. We discuss pros and cons of each method, error estimates, and document their performance with a few numerical experiments.

Abstract: Classical numerical methods for solving partial differential equations suffer from the curse dimensionality mainly due to their reliance on meticulously generated spatio-temporal grids. Inspired by modern deep learning based techniques for solving forward and inverse problems associated with partial differential equations, we circumvent the tyranny of numerical discretization by devising an algorithm that is scalable to high-dimensions. In particular, we approximate the unknown solution by a deep neural network which essentially enables us to benefit from the merits of automatic differentiation. To train the aforementioned neural network we leverage the well-known connection between high-dimensional partial differential equations and forward-backward stochastic differential equations. In fact, independent realizations of a standard Brownian motion will act as training data. We test the effectiveness of our approach for a couple of benchmark problems spanning a number of scientific domains including Black-Scholes-Barenblatt and Hamilton-Jacobi-Bellman equations, both in 100-dimensions.

Abstract: The free vibration analysis of a circular plate made up of a porous material integrated by piezoelectric actuator patches has been studied. The plate is assumed to be thin and its shear deformations have been neglected. The porous material properties vary through the plate thickness according to some given functions. Using Hamilton's variational principle and the classical plate theory (CPT) the governing motion equations have been obtained. Simple and clamped supports have been considered for the boundary conditions. The differential quadrature method (DQM) has been used for the discretizations required for numerical analysis. The effect of some parameters such as thickness ratio, porosity, piezoelectric actuators, variation of piezoelectric actuators-to-porous plate thickness ratio, pores distribution and pores compressibility on the natural frequency, radial and circumferential stresses has been illustrated. The results have been compared with the similar ones in the literature.

Abstract: In this article, by popularization of the reproducing kernel Hilbert space method in the sense of the Atangana–Baleanu fractional operator; set of first-order integrodifferential equations are solved with respect to Fredholm operator and initial conditions of optimality. The solvability approach based on use of the generalized Mittag–Leffler function in order to avoid nonsingular and nonlocal kernel functions appears in the classical fractional operator's. The procedure of solution is studied and described in details under some hypotheses, which provides the theoretical structure behind the utilized numerical method. Indeed, error analysis and convergence of numerical solution for the identification of the method is introduced in Hilbert space. In this analysis, some computational results and graphical representations are presented to demonstrated the suitability and portability of the utilized new fractional operator. Finally, the gained results reach to that; the utilized method is simple, direct, and powerful tool in finding numerical solutions for considered fractional equations.

Abstract: The aim of the present analysis is to implement a relatively recent computational algorithm, reproducing kernel Hilbert space, for obtaining the solutions of systems of first-order, two-point boundary value problems for ordinary differential equations. The reproducing kernel Hilbert space is constructed in which the initial–final conditions of the systems are satisfied. Whilst, three smooth kernel functions are used throughout the evolution of the algorithm in order to obtain the required grid points. An efficient construction is given to obtain the numerical solutions for the systems together with an existence proof of the exact solutions based upon the reproducing kernel theory. In this approach, computational results of some numerical examples are presented to illustrate the viability, simplicity, and applicability of the algorithm developed. Finally, the utilized results show that the present algorithm and simulated annealing provide a good scheduling methodology to such systems compared with other numerical methods.

Abstract: In this study, with the aid of Wolfram Mathematica 11, the modified exp ( − Ω ( η ) ) -expansion function method is used in constructing some new analytical solutions with novel structure such as the trigonometric and hyperbolic function solutions to the well-known nonlinear evolutionary equation, namely; the two-component second order KdV evolutionary system. Second, the finite forward difference method is used in analyzing the numerical behavior of this equation. We consider equation (6.5) and (6.6) for the numerical analysis. We examine the stability of the two-component second order KdV evolutionary system with the finite forward difference method by using the Fourier-Von Neumann analysis. We check the accuracy of the finite forward difference method with the help of L 2 and L ∞ norm error. We present the comparison between the exact and numerical solutions of the two-component second order KdV evolutionary system obtained in this article which and support with graphics plot. We observed that the modified exp ( − Ω ( η ) ) -expansion function method is a powerful approach for finding abundant solutions to various nonlinear models and also finite forward difference method is efficient for examining numerical behavior of different nonlinear models.

Abstract: Variable-order differential operators can be employed as a powerful tool to modeling nonlinear fractional differential equations and chaotical systems. In this paper, we propose a new generalize numerical schemes for simulating variable-order fractional differential operators with power-law, exponential-law and Mittag-Leffler kernel. The numerical schemes are based on the fundamental theorem of fractional calculus and the Lagrange polynomial interpolation. These schemes were applied to simulate the chaotic financial system and memcapacitor-based circuit chaotic oscillator. Numerical examples are presented to show the applicability and efficiency of this novel method.

Abstract: This article is about numerical methods for the solution of nonlinear equations. We consider both the fixed-point form and the equations form and explain why both versions are necessary to understand the solvers. We include the classical methods to make the presentation complete and discuss less familiar topics such as Anderson acceleration, semi-smooth Newton’s method, and pseudo-arclength and pseudo-transient continuation methods.

Abstract: Stochastic differential equations (SDEs) and the Kolmogorov partial differential equations (PDEs) associated to them have been widely used in models from engineering, finance, and the natural sciences. In particular, SDEs and Kolmogorov PDEs, respectively, are highly employed in models for the approximative pricing of financial derivatives. Kolmogorov PDEs and SDEs, respectively, can typically not be solved explicitly and it has been and still is an active topic of research to design and analyze numerical methods which are able to approximately solve Kolmogorov PDEs and SDEs, respectively. Nearly all approximation methods for Kolmogorov PDEs in the literature suffer under the curse of dimensionality or only provide approximations of the solution of the PDE at a single fixed space-time point. In this paper we derive and propose a numerical approximation method which aims to overcome both of the above mentioned drawbacks and intends to deliver a numerical approximation of the Kolmogorov PDE on an entire region $[a,b]^d$ without suffering from the curse of dimensionality. Numerical results on examples including the heat equation, the Black-Scholes model, the stochastic Lorenz equation, and the Heston model suggest that the proposed approximation algorithm is quite effective in high dimensions in terms of both accuracy and speed.

Abstract: A coupled thermo-mechanical bond-based peridynamical (TM-BB-PD) method is developed to simulate thermal cracking processes in rocks. The coupled thermo-mechanical model consists of two parts. In the first part, temperature distribution of the system is modeled based on the heat conduction equation. In the second part, the mechanical deformation caused by temperature change is calculated to investigate thermal fracture problems. The multi-rate explicit time integration scheme is proposed to overcome the multi-scale time problem in coupled thermo-mechanical systems. Two benchmark examples, i.e., steady-state heat conduction and transient heat conduction with deformation problem, are performed to illustrate the correctness and accuracy of the proposed coupled numerical method in dealing with thermo-mechanical problems. Moreover, two kinds of numerical convergence for peridynamics, i.e., m- and $$\delta $$ -convergences, are tested. The thermal cracking behaviors in rocks are also investigated using the proposed coupled numerical method. The present numerical results are in good agreement with the previous numerical and experimental data. Effects of PD material point distributions and nonlocal ratios on thermal cracking patterns are also studied. It can be found from the numerical results that thermal crack growth paths do not increases with changes of PD material point spacing when the nonlocal ratio is larger than 4. The present numerical results also indicate that thermal crack growth paths are slightly affected by the arrangements of PD material points. Moreover, influences of thermal expansion coefficients and inhomogeneous properties on thermal cracking patterns are investigated, and the corresponding thermal fracture mechanism is analyzed in simulations. Finally, a LdB granite specimen with a borehole in the heated experiment is taken as an application example to examine applicability and usefulness of the proposed numerical method. Numerical results are in good agreement with the previous experimental and numerical results. Meanwhile, it can be found from the numerical results that the coupled TM-BB-PD has the capacity to capture phenomena of temperature jumps across cracks, which cannot be captured in the previous numerical simulations.

Abstract: In this paper, an efficient numerical method is introduced for solving the fractional (Caputo sense) Fisher equation. This equation presents the problem of biological invasion and occurs, e.g., in ecology, physiology, and in general phase transition problems and others. We use the spectral collocation method which is based upon Chebyshev approximations. The properties of Chebyshev polynomials are utilized to reduce the proposed problem to a system of ODEs, which is solved by using finite difference method (FDM). Some theorems about the convergence analysis are stated. A numerical simulation and a comparison with the previous work are presented. We can apply the proposed method to solve other problems in engineering and physics.

Abstract: We study a finite element computational model for solving the coupled problem arising in the interaction between a free fluid and a fluid in a poroelastic medium. The free fluid is governed by the Stokes equations, while the flow in the poroelastic medium is modeled using the Biot poroelasticity system. Equilibrium and kinematic conditions are imposed on the interface. A mixed Darcy formulation is employed, resulting in continuity of flux condition of essential type. A Lagrange multiplier method is employed to impose weakly this condition. A stability and error analysis is performed for the semi-discrete continuous-in-time and the fully discrete formulations. A series of numerical experiments is presented to confirm the theoretical convergence rates and to study the applicability of the method to modeling physical phenomena and the robustness of the model with respect to its parameters.

Abstract: This paper presents a new computational technique for solving fractional pantograph differential equations. The fractional derivative is described in the Caputo sense. The main idea is to use Muntz-Legendre wavelet and its operational matrix of fractional-order integration. First, the Muntz-Legendre wavelet is presented. Then a family of piecewise functions is proposed, based on which the fractional order integration of the Muntz-Legendre wavelets are easy to calculate. The proposed approach is used this operational matrix with the collocation points to reduce the under study problem to a system of algebraic equations. An estimation of the error is given in the sense of Sobolev norms. The efficiency and accuracy of the proposed method are illustrated by several numerical examples.

Abstract: We provide both a general framework for discretizing de Rham sequences of differential forms of high regularity, and some examples of finite element spaces that fit in the framework. The general framework is an extension of the previously introduced notion of finite element systems, and the examples include conforming mixed finite elements for Stokes’ equation. In dimension 2 we detail four low order finite element complexes and one infinite family of highorder finite element complexes. In dimension 3 we define one low order complex, which may be branched into Whitney forms at a chosen index. Stokes pairs with continuous or discontinuous pressure are provided in arbitrary dimension. The finite element spaces all consist of composite polynomials. The framework guarantees some nice properties of the spaces, in particular the existence of commuting interpolators. It also shows that some of the examples are minimal spaces.

Abstract: We describe a strategy to solve differential equations for Feynman integrals by powers series expansions near singular points and to obtain high precision results for the corresponding master integrals. We consider Feynman integrals with two scales, i.e. non-trivially depending on one variable. The corresponding algorithm is oriented at situations where canonical form of the differential equations is impossible. We provide a computer code constructed with the help of our algorithm for a simple example of four-loop generalized sunset integrals with three equal non-zero masses and two zero masses. Our code gives values of the master integrals at any given point on the real axis with a required accuracy and a given order of expansion in the regularization parameter ϵ.

Abstract: The inverse problem of Kohn–Sham density functional theory (DFT) is often solved in an effort to benchmark and design approximate exchange-correlation potentials. The forward and inverse problems of DFT rely on the same equations but the numerical methods for solving each problem are substantially different. We examine both problems in this tutorial with a special emphasis on the algorithms and error analysis needed for solving the inverse problem. Two inversion methods based on partial differential equation constrained optimization and constrained variational ideas are introduced. We compare and contrast several different inversion methods applied to one-dimensional finite and periodic model systems.

Abstract: In the paper we present results of accuracy evaluation of numerous numerical algorithms for the numerical approximation of the Inverse Laplace Transform. The selected algorithms represent diverse lines of approach to this problem and include methods by Stehfest, Abate and Whitt, Vlach and Singhai, De Hoog, Talbot, Zakian and a one in which the FFT is applied for the Fourier series convergence acceleration. We use C++ and Python languages with arbitrary precision mathematical libraries to address some crucial issues of numerical implementation. The test set includes Laplace transforms considered as difficult to compute as well as some others commonly applied in fractional calculus. Evaluation results enable to conclude that the Talbot method which involves deformed Bromwich contour integration, the De Hoog and the Abate and Whitt methods using Fourier series expansion with accelerated convergence can be assumed as general purpose high-accuracy algorithms. They can be applied to a wide variety of inversion problems.

Abstract: The purpose of this paper is to continue to describe the development of a comprehensive methodology for fully resolved numerical simulations of fused deposition modeling.,A front-tracking/finite volume method introduced in Part I to simulate the heat transfer and fluid dynamics of the deposition of a polymer filament on a fixed bed is extended by adding an improved model for the injection nozzle, including the shrinkage of the polymer as it cools down, and accounting for stresses in the solid.,The accuracy and convergence properties of the new method are tested by grid refinement, and the method is shown to produce convergent solutions for the shape of the filament, the temperature distribution, the shrinkage and the solid stresses.,The method presented in the paper focuses on modeling the fluid flow, the cooling and solidification and volume changes and residual stresses, using a relatively simple viscoelastic constitutive model. More complex material models, depending, for example, on the evolution of the conformation tensor, are not included.,The ability to carry out fully resolved numerical simulations of the fused deposition process is expected to be critical for the validation of mathematical models for the material behavior, to help explore new deposition strategies and to provide the “ground truth” for the development of reduced-order models.,The paper completes describing the development of the first numerical method for fully resolved simulation of fused filament modeling.

Abstract: Entropy conservation and stability of numerical methods in gas dynamics have received much interest. Entropy conservative numerical fluxes can be used as ingredients in two kinds of schemes: firstly, as building blocks in the subcell flux differencing form of Fisher and Carpenter (Technical Report NASA/TM-2013-217971, NASA, 2013; J Comput Phys 252:518–557, 2013) and secondly (enhanced by dissipation) as numerical surface fluxes in finite volume like schemes. The purpose of this article is threefold. Firstly, the flux differencing theory is extended, guaranteeing high-order for general symmetric and consistent numerical fluxes and investigating entropy stability in a generalised framework of summation-by-parts operators applicable to multiple dimensions and simplex elements. Secondly, a general procedure to construct affordable entropy conservative fluxes is described explicitly and used to derive several new fluxes. Finally, robustness properties of entropy stable numerical fluxes are investigated and positivity preservation is proven for several entropy conservative fluxes enhanced with local Lax–Friedrichs type dissipation operators. All these theoretical investigations are supplemented with numerical experiments.

Abstract: For short fiber reinforced plastic parts the local fiber orientation has a strong influence on the mechanical properties. To enable multiscale computations using surrogate models we advocate a two-step identification strategy. Firstly, for a number of sample orientations an effective model is derived by numerical methods available in the literature. Secondly, to cover a general orientation state, these effective models are interpolated. In this article we develop a novel and effective strategy to carry out this interpolation. Firstly, taking into account symmetry arguments, we reduce the fiber orientation phase space to a triangle in $${\mathbb {R}}^2$$ . For an associated triangulation of this triangle we furnish each node with an surrogate model. Then, we use linear interpolation on the fiber orientation triangle to equip each fiber orientation state with an effective stress. The proposed approach is quite general, and works for any physically nonlinear constitutive law on the micro-scale, as long as surrogate models for single fiber orientation states can be extracted. To demonstrate the capabilities of our scheme we study the viscoelastic creep behavior of short glass fiber reinforced PA66, and use Schapery’s collocation method together with FFT-based computational homogenization to derive single orientation state effective models. We discuss the efficient implementation of our method, and present results of a component scale computation on a benchmark component by using ABAQUS ®.