Abstract: Quantum computers can produce a quantum encoding of the solution of a system of differential equations exponentially faster than a classical algorithm can produce an explicit description. However, while high-precision quantum algorithms for linear ordinary differential equations are well established, the best previous quantum algorithms for linear partial differential equations (PDEs) have complexity $\mathrm{poly}(1/\epsilon)$, where $\epsilon$ is the error tolerance. By developing quantum algorithms based on adaptive-order finite difference methods and spectral methods, we improve the complexity of quantum algorithms for linear PDEs to be $\mathrm{poly}(d, \log(1/\epsilon))$, where $d$ is the spatial dimension. Our algorithms apply high-precision quantum linear system algorithms to systems whose condition numbers and approximation errors we bound. We develop a finite difference algorithm for the Poisson equation and a spectral algorithm for more general second-order elliptic equations.

Abstract: In this work, we introduce a numerical and analytical study of the Peyrard-Bishop DNA dynamic model equation. This model is studied analytically by hyperbolic and exponential ansatz methods and numerically by finite difference method. A comparison between the results obtained by the analytical methods and the numerical method is investigated. Furthermore, some figures are introduced to show how accurate the solutions will be obtained from the analytical and numerical methods.

Abstract: The theory of mean field games aims at studying deterministic or stochastic differential games (Nash equilibria) as the number of agents tends to infinity. Since very few mean field games have explicit or semi-explicit solutions, numerical simulations play a crucial role in obtaining quantitative information from this class of models. They may lead to systems of evolutive partial differential equations coupling a backward Bellman equation and a forward Fokker–Planck equation. In the present survey, we focus on such systems. The forward-backward structure is an important feature of this system, which makes it necessary to design unusual strategies for mathematical analysis and numerical approximation. In this survey, several aspects of a finite difference method used to approximate the previously mentioned system of PDEs are discussed, including convergence, variational aspects and algorithms for solving the resulting systems of nonlinear equations. Finally, we discuss in details two applications of mean field games to the study of crowd motion and to macroeconomics, a comparison with mean field type control, and present numerical simulations.

Abstract: In this paper, a three-dimensional (3D) cascaded lattice Boltzmann method (CLBM) is implemented to simulate the liquid–vapor phase-change process. The multiphase flow field is solved by incorporating the pseudopotential multiphase model into an improved CLBM, the temperature field is solved by the finite difference method, and the two fields are coupled via a non-ideal equation of state. Through numerical simulations of several canonical problems, it is verified that the proposed phase-change CLBM is applicable for both the isothermal multiphase flow and the liquid–vapor phase-change process. Using the developed method, a complete 3D pool boiling process with up to hundreds of spontaneously generated bubbles is simulated, faithfully reproducing the nucleate boiling, transition boiling, and film boiling regimes. It is shown that the critical heat flux predicted by the 3D simulations agrees better with the established theories and correlation equations than that obtained by two-dimensional simulations. Furthermore, it is found that with the increase in the wall superheats, the bubble footprint area distribution changes from an exponential distribution to a power-law distribution, in agreement with experimental observations. In addition, insights into the instantaneous and time-averaged characteristics of the first two largest bubble footprints are obtained.

Abstract: The present article develops a semi-discrete numerical scheme to solve the time-fractional stochastic advection–diffusion equations. This method, which is based on finite difference scheme and radial basis functions (RBFs) interpolation, is applied to convert the solution of time-fractional stochastic advection–diffusion equations to the solution of a linear system of algebraic equations. The mechanism of this method is such that time-fractional stochastic advection–diffusion equation is first transformed into elliptic stochastic differential equations by using finite difference scheme. Then meshfree method based on RBFs has been used to approximate the resulting equation. In other words, the approximate solution of time-fractional stochastic advection–diffusion equation is achieved with discrete the domain in the t-direction by finite difference method and approximating the unknown function in the x-direction by generalized inverse multiquadrics RBFs. In this method, the noise terms are directly simulated at the collocation points in each time step and it is the most important advantage of the suggested approach. Stability and convergence of the scheme are established. Finally, some test problems are included to confirm the accuracy and efficiency of the new approach.

Abstract: In this paper, we present a time two-grid algorithm based on the finite difference (FD) method for the two-dimensional nonlinear time-fractional mobile/immobile transport model. We establish the problem as a nonlinear fully discrete FD system, where the time derivative is discretized by the second-order backward difference formula (BDF) scheme, the Caputo fractional derivative is treated by means of L1 discretization formula, and the spatial derivative is approximated by the central difference formula. For solving the nonlinear FD system more efficiently, a time two-grid algorithm is proposed, which consists of two steps: first, the nonlinear FD system on a coarse grid is solved by nonlinear iterations; second, the Newton iteration is utilized to solve the linearized FD system on the fine grid. The stability and convergence in L2-norm are obtained for the two-grid FD scheme. Numerical results are consistent with the theoretical analysis. Meanwhile, numerical experiments show that the two-grid FD method is much more efficient than the general FD scheme for solving the nonlinear FD system.

Abstract: In this article, a fractional-order mathematical physics model, advection–dispersion equation (FADE), will be solved numerically through a new approximative technique. Shifted Vieta–Lucas orthogonal polynomials will be considered as the main base for the desired numerical solution. These polynomials are used for transforming the FADE into an ordinary differential equations system (ODES). The nonstandard finite difference method coincidence with the spectral collocation method will be used for converting the ODES into an equivalence system of algebraic equations that can be solved numerically. The Caputo fractional derivative will be used. Moreover, the error analysis and the upper bound of the derived formula error will be investigated. Lastly, the accuracy and efficiency of the proposed method will be demonstrated through some numerical applications.

Abstract: In this paper, we study finite difference methods for fractional differential equations (FDEs) with Caputo–Hadamard derivatives. First, smoothness properties of the solution are investigated. The fractional rectangular, $${L}_{\mathrm{log},1}$$ interpolation, and modified predictor–corrector methods for Caputo–Hadamard fractional ordinary differential equations (FODEs) are proposed through approximating the corresponding equivalent Volterra integral equations. The stability and error estimate of the derived methods are proved as well. Then, we investigate finite difference methods for fractional partial differential equations (FPDEs) with Caputo–Hadamard derivative. By applying the constructed L1 scheme for approximating the time fractional derivative, a semi-discrete difference scheme is derived. The stability and convergence analysis are shown too in detail. Furthermore, a fully discrete scheme is established by the standard second-order difference scheme in spacial direction. Stability and error estimate are also presented. The numerical experiments are displayed to verify the theoretical results.

Abstract: In the present work, a generalized finite difference method (GFDM), a meshless method based on Taylor-series approximations, is proposed to solve stationary 2D and 3D Stokes equations. To overcome the troublesome pressure oscillation in the Stokes problem, a new simple formulation of boundary condition for the Stokes problem is proposed. This numerical approach only adds a mixed boundary condition, the projections of the momentum equation on the boundary outward normal vector, to the Stokes equations, without any other change to the governing equations. The proposed formulation can be easily discretized by the GFDM. The GFDM is evolved from the Taylor series expansions and moving-least squares approximation, and the derivative expressed of unknown variables as linear combinations of function values of neighboring nodes. Numerical examples are utilized to verify the feasibility of the proposed GFDM scheme not only for the Stokes problem, but also for more involved and general problems, such as the Poiseuille flow, the Couette flow and the Navier–Stokes equations in low-Reynolds-number regime. Moreover, numerical results and comparisons show that using the GFDM to solve the proposed formulation of the Stokes equations is more accurate than the classical formulation of the pressure Poisson equation.

Abstract: In this paper, we study an accurate numerical method for solving the multi space-fractional Gardner equation (MSFGE) with the Caputo–Fabrizio (CF) and Atangana–Baleanu (AB) fractional derivatives where the space-fractional terms are under the sense of Caputo. To the best knowledge of the reader, we are first to use the spectral collocation method based on the third Chebyshev approximations to reduced the multi space-fractional Gardner equation to a system of ordinary differential equations by using the properties of Chebyshev polynomials and then solved them via the finite difference method (FDM). By computing the absolute errors we present the effectiveness and accuracy of the proposed methods. The present paper investigates the dynamics of Gardner Equation by considering two fractional operators that is the Caputo–Fabrizio and Atangana–Baleanu, this is entirely new idea by using the two operators on a Gardner equation. Our results prove that the given procedure is an easy and efficient tool to investigate the solution of nonlinear equations with local and non-local singular kernels.

Abstract: The exact solution is calculated for fractional telegraph partial differential equation depend on initial boundary value problem. Stability estimates are obtained for this equation. Crank-Nicholson difference schemes are constructed for this problem. The stability of difference schemes for this problem is presented. This technique has been applied to deal with fractional telegraph differential equation defined by Caputo fractional der

Abstract: This paper presents results of comparison of modern effective numerical techniques for the calculation of microwave devices. Nowadays finite difference time domain method and finite elements method are the most frequently used numerical techniques, which are applied for the simulation of electromagnetic fields in various antennas and bounded structures. Modern modifications of these numerical methods and their pro and contras are considered in the paper. Besides, we have compared finite difference time domain technique and finite elements method in the case of calculation of polarization and matching characteristics of microwave waveguide iris polarizers. Using this example we have found that the convergence of the calculated matching characteristics of microwave waveguide devices is fast for both considered numerical techniques. On the other hand, the accuracy of calculation of the phase and polarization characteristics is very sensitive to the number of mesh cells, at which the volume of device structure is divided. It has been found that more than 100 000 tetrahedral mesh cells per structure volume must be used, if calculation of the polarizer’s axial ratio and crosspolar discrimination is performed by finite elements method in the frequency domain with required accuracies of 0.5 dB. If axial ratio and crosspolar discrimination must be calculated with the accuracies of 0.5 dB by the finite difference time domain method, then the utilization of more than 800 000 hexahedral mesh cells per structure volume is required. In has been found that the computation time of the finite elements method in the frequency domain is more than 2 times less than the same time required by finite difference time domain method. Besides, the corresponding number of tetrahedral mesh cells in finite elements method is 10 times less, than the number of hexahedral mesh cells in finite difference time domain method.

Abstract: There are many unicellular tiny organisms which can self-propel collectively through non-Newtonian fluids by means of producing undulating deformation. Example includes nematodes, rod shaped bacteria and spermatozoa. Here we use Taylor’s swimming sheet model, with non-Newtonian fluid bounded with in a complex wavy walls of a two-dimensional channel. Oldroyd-4 constant fluid is approximated as cervical mucus and MHD effects are also considered. After utilizing lubrication and creeping flow assumption the reduced non-linear differential equation is solved (by implicit finite difference technique) so that it will satisfy the dynamic equilibrium condition for steady propulsion. For a special (Newtonian) case the expressions of swimming speed and flow rate are also presented. We also demonstrate that the rheological properties of non-Newtonian fluid can assist or resist the pack of micro-organisms (swimming sheet), while the larger undulation amplitude in swimmer’s body and magnetic field in downward direction can enhance the propulsion speed. The solution obtained via implicit finite difference method is also validated by a built in MATLAB routine bvp-4c. This built in function is based on collocation technique. Moreover an excellent correlation is achieved for both numerical methods.

Abstract: The idea of fractional derivatives is applied to several problems of viscoelastic fluid. However, most of these problems (fluid problems), were studied analytically using different integral transform techniques, as most of these problems are linear. The idea of the above fractional derivatives is rarely applied to fluid problems governed by nonlinear partial differential equations. Most importantly, in the nonlinear problems, either the fractional models are developed by artificial replacement of the classical derivatives with fractional derivatives or simple classical problems (without developing the fractional model even using artificial replacement) are solved. These problems were mostly solved for steady-state fluid problems. In the present article, studied unsteady nonlinear non-Newtonian fluid problem (Cattaneo-Friedrich Maxwell (CFM) model) and the fractional model are developed starting from the fractional constitutive equations to the fractional governing equations; in other words, the artificial replacement of the classical derivatives with fractional derivatives is not done, but in details, the fractional problem is modeled from the fractional constitutive equations. More exactly two-dimensional magnetic resistive flow in a porous medium of fractional Maxwell fluid (FMF) over an inclined plate with variable velocity and the temperature is studied. The Caputo time-fractional derivative model (CFM) is used in the governing equations. The proposed model is numerically solved via finite difference method (FDM) along with L1-scheme for discretization. The numerical results are presented in various figures. These results indicated that the fractional parameters significantly affect the temperature and velocity fields. It is noticed that the temperature field increased with an increase in the fractional parameter. Whereas, the effect of fractional parameters is opposite on the velocity field near the plate. However, this trend became like that of the temperature profile, away from the plate. Moreover, the velocity field retarded with strengthening in the magnetic parameter due to enhancement in Lorentz force. However, this effect reverses in the case of the temperature profile.

Abstract: In this work, we have developed a numerical algorithm based on redefined extended cubic B-spline functions to explore the approximate solution for time fractional Klein--Gorden equation. The presented technique engages finite difference formulation for discretizing the Caputo time fractional derivative of order $\alpha \in (1,2]$ and redefined extended cubic B-spline functions to interpolate the solution curve along spatial grid. A stability analysis of the scheme is set up to affirm that the errors do not amplify during the execution of numerical procedure. The derivation of uniform convergence reveals that the scheme is $O(h^{2}+\Delta t^{2-\alpha})$ accurate. Some computational experiments are carried out to verify the theoretical considerations. Numerical results are compared with the existing techniques on the topic and it is concluded that the present scheme returns superior outcomes.

Abstract: Abstract An initial-boundary value problem, whose differential equation contains a sum of fractional time derivatives with orders between 0 and 1, is considered. Its spatial domain is ( 0 , 1 ) d {(0,1)^{d}} for some d ∈ { 1 , 2 , 3 } {d\\in\\{1,2,3\\}} . This problem is a generalisation of the problem considered by Stynes, O’Riordan and Gracia in SIAM J. Numer. Anal. 55 (2017), pp. 1057–1079, where d = 1 {d=1} and only one fractional time derivative was present. A priori bounds on the derivatives of the unknown solution are derived. A finite difference method, using the well-known L1 scheme for the discretisation of each temporal fractional derivative and classical finite differences for the spatial discretisation, is constructed on a mesh that is uniform in space and arbitrarily graded in time. Stability and consistency of the method and a sharp convergence result are proved; hence it is clear how to choose the temporal mesh grading in a optimal way. Numerical results supporting our theoretical results are provided.

Abstract: Recent years have witnessed growing interests in solving partial differential equations by deep neural networks, especially in the high-dimensional case. Unlike classical numerical methods, such as finite difference method and finite element method, the enforcement of boundary conditions in deep neural networks is highly nontrivial. One general strategy is to use the penalty method. In the work, we conduct a comparison study for elliptic problems with four different boundary conditions, i.e., Dirichlet, Neumann, Robin, and periodic boundary conditions, using two representative methods: deep Galerkin method and deep Ritz method. In the former, the PDE residual is minimized in the least-squares sense while the corresponding variational problem is minimized in the latter. Therefore, it is reasonably expected that deep Galerkin method works better for smooth solutions while deep Ritz method works better for low-regularity solutions. However, by a number of examples, we observe that deep Ritz method can outperform deep Galerkin method with a clear dependence of dimensionality even for smooth solutions and deep Galerkin method can also outperform deep Ritz method for low-regularity solutions. Besides, in some cases, when the boundary condition can be implemented in an exact manner, we find that such a strategy not only provides a better approximate solution but also facilitates the training process.