Dimensional Partial Differential Equations Research Articles

Simulations of the one and two dimensional nonlinear evolutionary partial differential equations: A numerical study

In this work a hybrid scheme is proposed for the numerical study of various evolutionary partial differential equations (EPDEs). In proposed strategy, temporal derivatives are estimated via first and second order finite differences while the solution and spatial derivatives are approximated by Lucas and Fibonacci polynomials. Further, the collocation approach is applied which convert the EPDEs, to the system of coupled linear equations which are simple to solve. For non-linear problems, Quasilinearization is used to tackle nonlinearity. The scheme is implemented to solve different EPDEs which include, one dimensional non-linear Boussinesq, Hunter–Saxton, wave-like and two dimensional linear and non-linear wave like equations. Accuracy of the scheme is portrayed by computing L∞,L2 error norms and the relative error. Moreover, the calculated outcomes are compared with previously existing results in literature. Simulation demonstrates that the scheme works well for the mentioned EPDEs.

Special Functions and Its Application in Solving Two Dimensional Hyperbolic Partial Differential Equation of Telegraph Type

In this article, we use the applications of special functions in the form of Chebyshev polynomials to find the approximate solution of hyperbolic partial differential equations (PDEs) arising in the mathematical modeling of transmission line subject to appropriate symmetric Dirichlet and Neumann boundary conditions. The special part of the model equation is discretized using a Chebyshev differentiation matrix, which is centro-asymmetric using the symmetric collocation points as grid points, while the time derivative is discretized using the standard central finite difference scheme. One of the disadvantages of the Chebyshev differentiation matrix is that the resultant matrix, which is obtained after replacing the special coordinates with the derivative of Chebyshev polynomials, is dense and, therefore, needs more computational time to evaluate the resultant algebraic equation. To overcome this difficulty, an algorithm consisting of fast Fourier transformation is used. The main advantage of this transformation is that it significantly reduces the computational cost needed for N collocation points. It is shown that the proposed scheme converges exponentially, provided the data are smooth in the given equations. A number of numerical experiments are performed for different time steps and compared with the analytical solution, which further validates the accuracy of our proposed scheme.

Homotopy analysis of mixed convection flow of a magnetized viscoelastic nanofluid from a stretching surface in non-Darcy porous media with revised Fourier and Fickian approaches

This article addresses theoretically the mixed convection hydromagnetic flow of electrically conducting viscoelastic nanofluid from a vertical permeable stretching sheet in a non-Darcy porous medium with Cattaneo–Christov double-diffusion model to evaluate the heat transfer phenomena in steady boundary layer flow on a stretchable surface. In this regard, thermal and solutal hyperbolic wave relaxation effects are included for non-Fourier and non-Fickian models. Heat absorption is also included in the analysis. The Buongiorno two-component nanoscale model is adopted for simulating Brownian motion and thermophoretic body force effects. A non-Darcy drag force model is employed for the porous medium and the Reiner–Rivlin second-grade model for non-Newtonian characteristics. Via appropriate dimensionless similarity variables, the nonlinear dimensional partial differential conservation equations for momentum, energy and concentration with associated boundary conditions are rendered into a nonlinear dimensionless ordinary differential boundary value problem. The homotopy analysis method (HAM) is utilized to solve the boundary value problem and the impact of emerging parameters including thermal relaxation parameters, non-Newtonian material parameter, Darcy permeability parameter, porous inertial parameter and Hartmann magnetic number on the momentum, heat and mass transfer characteristics are visualized graphically and in tables. A 30th-order approximation for HAM is shown to produce sufficient accuracy for the velocity and temperature fields and a 40th-order estimates is adequate for the concentration field. It is observed that increasing the magnitude of thermal and solutal relaxation parameters has the opposite effect on the thermal and concentration distributions. The novelty of the work is the simultaneous inclusion of multiple effects (non-Fourier and non-Fickian thermal relaxation hyperbolic wave models, non-Darcy drag effects and heat generation/absorption) which are relevant to rheological nanomaterials processing and also the deployment of homotopy analysis as an alternative to conventional numerical methods such as finite differences, finite elements and MATLAB solvers. The study is relevant to the manufacture of electro-conductive polymers (ECPs) and smart (functional) magnetic nano-liquids.

MHD flow of nanofluid over moving slender needle with nanoparticles aggregation and viscous dissipation effects.

The study of boundary layer flows over an irregularly shaped needle with small horizontal and vertical dimensions is popular among academics because it seems to have a lot of uses in fields as different as bioinformatics, medicine, engineering, and aerodynamics. With nanoparticle aggregation, magnetohydrodynamics, and viscous dissipation all playing a role in the flow and heat transmission of an axisymmetric nanofluid via a moving thin needle, this article provides guidance on how to employ a boundary layer for this purpose. In this case, we utilized the similarity transformation to change the dimensional partial differential equation into the dimensionless ordinary differential equation. We utilize MATHEMATICA to include shooting using RK-IV methods after identifying the numerical issue. Several characteristics were measured, leading to the discovery of a broad variety of values for things like skin friction coefficients, Nusselt numbers, velocity profiles, and temperature distributions. Velocity profile decreases with increasing values of and increases against Temperature profiles enhances with increasing values of , and The reduction in skin friction between a needle and a fluid can be observed when the values of M and are boosted. Furthermore, it was also noticed an increase in heat transfer on needle surface dramatically when , and M were raised, whereas displayed the opposite effect. The findings of the current study are compared with prior findings for a particular instance in order to confirm the findings. Excellent agreement between the two sets of results is found.

New soliton solutions of modified (3+1)-D Wazwaz–Benjamin–Bona–Mahony and (2+1)-D cubic Klein–Gordon equations using first integral method

Abstract In this article, first integral method (FIM) is used to acquire the analytical solutions of (3+1)-D Wazwaz–Benjamin–Bona–Mahony and (2+1)-D cubic Klein–Gordon equation. New soliton solutions are obtained, such as solitons, cuspon, and periodic solutions. FIM is a direct method to acquire soliton solutions of nonlinear partial differential equations (PDEs). The proposed technique can be used for solving higher dimensional PDEs. FIM can be implemented to solve integrable and ion-integrable equations.

Greedy training algorithms for neural networks and applications to PDEs

Recently, neural networks have been widely applied for solving partial differential equations (PDEs). Although such methods have been proven remarkably successful on practical engineering problems, they have not been shown, theoretically or empirically, to converge to the underlying PDE solution with arbitrarily high accuracy. The primary difficulty lies in solving the highly non-convex optimization problems resulting from the neural network discretization, which are difficult to treat both theoretically and practically. It is our goal in this work to take a step toward remedying this. For this purpose, we develop a novel greedy training algorithm for shallow neural networks. Our method is applicable to both the variational formulation of the PDE and also to the residual minimization formulation pioneered by physics informed neural networks (PINNs). We analyze the method and obtain a priori error bounds when solving PDEs from the function class defined by shallow networks, which rigorously establishes the convergence of the method as the network size increases. Finally, we test the algorithm on several benchmark examples, including high dimensional PDEs, to confirm the theoretical convergence rate. Although the method is expensive relative to traditional approaches such as finite element methods, we view this work as a proof of concept for neural network-based methods, which shows that numerical methods based upon neural networks can be shown to rigorously converge.

Fibonacci wavelets-based numerical method for solving fractional order (1 + 1)-dimensional dispersive partial differential equation

Fibonacci wavelets-based numerical method for solving fractional order (1 + 1)-dimensional dispersive partial differential equation

Radiative and viscid dissipative flowing influences on heat and mass transfer in MHD Casson fluid employing Galerkin finite element style

This work employs a numerical calculation to investigate the heat and mass transfer process of a hydromagnetic radiative Casson fluid flowing through a vertical plate in the presence of heat generation, viscid dissipation and chemical processes. The governing dimensional partial differential equations (PDE) for considered Casson fluid are first converted into dimensionless PDE utilizing proper variables. The obtained system is then computed using Galerkin finite element method (G-FEM). The approximated results for different parameters are shown through curves and tables. The impact of dimensionless parameters is visualized on fluid velocity, fluid temperature, fluid concentration, Nusselt number and skin friction. The heat transmission mechanism of Casson fluid is superior to that of the Newtonian liquid. In the limited scenario, the study findings are in excellent accord with the existing literature.

On Some Finite Difference Schemes for the Solutions of Parabolic Partial Differential Equations

This paper presents the comparison of three different and unique finite difference schemes used for finding the solutions of parabolic partial differential equations (PPDE). Knowing fully that the efficiency of a numerical schemes depends solely on their stability therefore, the schemes were compared based on their stability using von Newmann method. The implicit scheme and Dufort-Frankel schemes using von Newmann stability method are unconditionally stable, while the explicit scheme is conditionally stable. The schemes were also applied to solve a one dimensional parabolic partial differential equations (heat equation) numerically and their results compared for best in efficiency. The numerical experiments as seen in the tables presented and also the percentage errors, which proves that the implicit scheme is good compare to the other two schemes. Also, the implementation of the implicit scheme is faster than that of the explicit and Dufort-Frankel schemes. The results obtained in work also compliment and agrees with the results in literature.

Spectral Analysis of High Order Continuous FEM for Hyperbolic PDEs on Triangular Meshes: Influence of Approximation, Stabilization, and Time-Stepping

In this work we study various continuous finite element discretization for two dimensional hyperbolic partial differential equations, varying the polynomial space (Lagrangian on equispaced, Lagrangian on quadrature points (Cubature) and Bernstein), the stabilization techniques (streamline-upwind Petrov–Galerkin, continuous interior penalty, orthogonal subscale stabilization) and the time discretization (Runge–Kutta (RK), strong stability preserving RK and deferred correction). This is an extension of the one dimensional study by Michel et al. (J Sci Comput 89(2):31, 2021. https://doi.org/10.1007/s10915-021-01632-7), whose results do not hold in multi-dimensional frameworks. The study ranks these schemes based on efficiency (most of them are mass-matrix free), stability and dispersion error, providing the best CFL and stabilization coefficients. The challenges in two-dimensions are related to the Fourier analysis. Here, we perform it on two types of periodic triangular meshes varying the angle of the advection, and we combine all the results for a general stability analysis. Furthermore, we introduce additional high order viscosity to stabilize the discontinuities, in order to show how to use these methods for tests of practical interest. All the theoretical results are thoroughly validated numerically both on linear and non-linear problems, and error-CPU time curves are provided. Our final conclusions suggest that Cubature elements combined with SSPRK and OSS stabilization is the most promising combination.

A hybrid collocation method for the computational study of multi-term time fractional partial differential equations

In this article, we focus on the numerical study of one and two dimensional higher order multi-term time fractional partial differential equations. In the adopted strategy, the temporal fractional derivative is replaced via well known L 1 formula and the integer order space derivatives are approximated by truncated one and two dimensional wavelet series. The fascinating nature of the scheme is to use collocation approach which convert the governing equations to the system of algebraic equations from which the wavelet coefficients can be calculated. Next, stability of the proposed scheme is investigated theoretically which is also the fundamental subject of the current work. Further computational convergence rate is computed which predicts that the order is approximately two. The scheme is applied to solve one dimensional beam models (fourth order partial differential equations) and two dimensional fissured rock models (Sobolev equations). Efficiency of the scheme is examined with the help of various error norms such as I ∞ , I r m s and I 2 . Simulations indicate pretty much good results from which we can say that the scheme is suitable and robust for both one and two dimensional problems.

Lie symmetry analysis, optimal system and exact solutions of variable-coefficients Sakovich equation

In this paper, the Sakovich equation is extended for the first time to a new equation with variable-coefficients on the time variable. The infinitesimal generators are obtained by performing a Lie symmetry analysis of the equation. Following this, the optimal system of one-dimensional subalgebras of the equation is analyzed. The (2+1)-dimensional variable-coefficients Sakovich equation is then reduced to (1+1)-dimensional partial differential equations by similarity reductions. The reduced partial differential equations are simplified to ordinary differential equations by the traveling wave transform. The new periodic wave solution and two-soliton interaction solution are obtained, and some exact solutions are obtained using the extended tanh-function method and the extended sech-function method. Finally, different types of soliton solutions are obtained by assigning different functions to the coefficient functions, and 3-dimensional figures are used to visualize the structural features of the soliton solutions.

Numerical computing of Soret and linear radiative effects on MHD Casson fluid flow toward a vertical surface through a porous medium: Finite element analysis

In this work, we investigate the time-dependent MHD free convection of Casson fluid across a vertical semi-infinite plate fitted inside a permeable medium, along with viscous dissipation, radiation absorption, and Soret effect by using several non-dimensional variables. The characteristics of a variety of elements influencing the flow phenomenon are examined using the Casson fluid model. The governing dimensional partial differential equations are transformed into an ordinary differential equation set by introducing the similarity variables. The reduced model is numerically solved via the Galerkin finite element method. The non-dimensional equations with suitable boundary conditions can be mathematically simplified using the efficient Galerkin finite element approach. The restrictions are shown numerically and graphically, and their effects on temperature, velocity, species concentration, and rate coefficients are all shown. This study is to present the influence of radiation absorption along with viscous dissipation on the heat transfer phenomenon. For different flow parameter estimations, graphs are generated for various flow profiles as well as skin friction coefficients. The Nusselt (Nu) and Sherwood (Sh) quantities are also demonstrated via graphs.

FRACTIONAL COMPLEX TRANSFORMS, REDUCED EQUATIONS AND EXACT SOLUTIONS OF THE FRACTIONAL KRAENKEL–MANNA–MERLE SYSTEM

Exact solutions of the fractional Kraenkel–Manna–Merle system in saturated ferromagnetic materials have been studied. Using the fractional complex transforms, the fractional Kraenkel–Manna–Merle system is reduced to ordinary differential equations, (1 + 1)-dimensional partial differential equations and (2 + 1)-dimensional partial differential equations. Based on the obtained ordinary differential equations and taking advantage of the available solutions of Jacobi elliptic equation and Riccati equation, soliton solutions, combined soliton solutions, combined Jacobi elliptic function solutions, triangular periodic solutions and rational function solutions, for the KMM system are obtained. For the obtained (1 + 1)-dimensional partial differential equations, we get the classification of Lie symmetries. Starting from a Lie symmetry, we get a symmetry reduction equation. Solving the symmetry reduction equation by the power series method, power series solutions for the KMM system are obtained. For the obtained (2 + 1)-dimensional partial differential equations, we derive their bilinear form and two-soliton solution. The bilinear form can also be used to study the lump solutions, rogue wave solutions and breathing wave solutions.

Lie symmetry analysis, exact solutions, optimal system and conservation laws for variable-coefficients Kundu–Mukherjee–Nasker equation

This paper investigates the (2+1)-dimensional variable coefficients Kundu–Mukherjee–Nasker equation which obtained by some transformations. The infinitesimal generators of the equation are obtained by the Lie group method. Next, the representative elements of the one-dimensional subalgebras optimal system are determined with adjoint representation. Based on the optimal system, (1+1)-dimensional partial differential equations can be obtained by similarity reductions. With the help of [Formula: see text]-expansion method, several exact solutions including hyperbolic function solutions, trigonometric function solutions and rational function solutions of variable coefficients Kundu–Mukherjee–Nasker equation are obtained. Finally, the conservation laws of the variable coefficients Kundu–Mukherjee–Nasker equation can be obtained by using the conservation theorem proposed by Ibragimov.

Computational analysis and heat transfer on MHD transient‐free convection flow in a vertical microporous channel in the existence of Hall and ion slip current

AbstractThis investigation presents an analysis of the transient magnetohydrodynamic flow of Newtonian viscous fluid in a vertical microporous channel with the inclusion of ion slip‐Hall current as well as an induced magnetic field (IMF) effects. Owing to the nature of the flow equations which are difficult to obtain an analytical result, a numerical scheme (PDEPE) based on finite difference approximation is adopted in solving the governing dimensional partial differential equation. The active influence of different flow parameters on velocity and IMF along the main flow and induced flow directions are visualized. Furthermore, variations of shear stress and induced current density are also presented in tabular form for active nondimensional flow quantities and later on analyzed. To establish the validity of the results obtained in this computation, values for velocity and IMF in this analysis were correlated with the steady‐state existing benchmark when the values of nondimensional time are considered large. Significant results from the analysis show that at the transient time and in the simultaneous occurrence of suction/injection at the microchannel walls, higher values of ion slip current have no considerable influence on flow formation along the primary flow direction, whereas an oscillatory phenomenon is observed along the secondary flow direction. It is also significant to note that at a transient time, magnetic induction could be improved or controlled by choosing favorable values of the suction/injection parameter.

Discrete gradient flow approximations of high dimensional evolution partial differential equations via deep neural networks

We consider the approximation of initial/boundary value problems involving, possibly high-dimensional, dissipative evolution partial differential equations (PDEs) using a deep neural network framework. More specifically, we first propose discrete gradient flow approximations based on non-standard Dirichlet energies for problems involving essential boundary conditions posed on bounded spatial domains. The imposition of the boundary conditions is realized weakly via non-standard functionals; the latter classically arise in the construction of Galerkin-type numerical methods and are often referred to as “Nitsche-type” methods. Moreover, inspired by the seminal work of Jordan, Kinderlehrer, and Otto (JKO) Jordan et al. (1998), we consider the second class of discrete gradient flows for special classes of dissipative evolution PDE problems with non-essential boundary conditions. These JKO-type gradient flows are solved via deep neural network approximations. A key, distinct aspect of the proposed methods is that the discretization is constructed via a sequence of residual-type deep neural networks (DNN) corresponding to implicit time-stepping. As a result, a DNN represents the PDE problem solution at each time node. This approach offers several advantages in the training of each DNN. We present a series of numerical experiments which showcase the good performance of Dirichlet-type energy approximations for lower space dimensions and the excellent performance of the JKO-type energies for higher spatial dimensions.

Time complexity analysis of quantum difference methods for linear high dimensional and multiscale partial differential equations

We investigate time complexities of finite difference methods for solving the high-dimensional linear heat equation, the high-dimensional linear hyperbolic equation and the multiscale hyperbolic heat system with quantum algorithms (hence referred to as the “quantum difference methods”). For the heat and linear hyperbolic equations we study the impact of explicit and implicit time discretizations on quantum advantages over the classical difference method. For the multiscale problem, we find the time complexity of both the classical treatment and quantum treatment for the explicit scheme scales as O(1/ε), where ε is the scaling parameter, while the scaling for the multiscale Asymptotic-Preserving (AP) schemes does not depend on ε. This indicates that it is still of great importance to develop AP schemes for multiscale problems in quantum computing.

Monte Carlo fPINNs: Deep learning method for forward and inverse problems involving high dimensional fractional partial differential equations

We introduce a sampling-based machine learning approach, Monte Carlo fractional physics-informed neural networks (MC-fPINNs), for solving forward and inverse fractional partial differential equations (FPDEs). As a generalization of the physics-informed neural networks (PINNs), MC-fPINNs utilize a Monte Carlo approximation strategy to compute the fractional derivatives of the DNN outputs, and construct an unbiased estimation of the physical soft constraints in the loss function. Our sampling approach can yield lower overall computational cost compared to fPINNs proposed in Pang et al.(2019), hence it can solve high dimensional FPDEs at reasonable cost. We validate the performance of MC-fPINNs via several examples, including high dimensional integral fractional Laplacian equations, parametric identification of time–space fractional PDEs, and fractional diffusion equation with random inputs. The results show that MC-fPINNs are flexible and quite effective in tackling high dimensional FPDEs.

A novel Fourier-based meshless method for [formula omitted]-dimensional fractional partial differential equation with general time-dependent boundary conditions

The article presents the novel Fourier-based meshless technique for solving (3+1)-dimensional fractional partial differential equation with coefficients varying in time under time-dependent boundary conditions of the general form. We transform the original equation into the one with homogeneous boundary conditions using a smooth analytical function and apply the Fourier expansion over the system of eigenfunctions of the Laplace operator corresponding to the zero boundary conditions which are solved independently using the semi-analytical backward substitution technique with the Müntz polynomial basis. The proposed method also can be used to solve problems of the integer orders or stationary ones where the Fourier expansion technique is suitable. The accuracy and efficiency of the mentioned procedure are demonstrated by solving high orders fractional equations.