Dimensional PDE Research Articles

Numerical approach for solving two dimensional fractal-fractional PDEs using peridynamic method

This paper develops a numerical approach for solving two-dimensional fractal-fractional parabolic partial differential equations. The fractal-fractional derivative is defined in the Atangana-Riemann-Liouville sense with Mittage-Leffler kernel. To solve this equation, we first eliminate the spatial derivatives using peridynamic differential operators. Then, we derive an operational matrix (OM) of fractal-fractional derivative in terms of the Legendre polynomials to simplify the time derivative. The aim of the formulated method is to transform the original problem into an uncomplicated system of linear algebraic equations which can be solved by mathematical software. The applicability of the approach is examined for several examples and numerical results show the computational efficiency of the method.

New Approach for Solving (2+1)-Dimensional Differential Equation

This article presents an exact analysis solution for (2+1) dimensional differential equations by using new approach based on coupled method via LA-transform with decomposition method to overcome the computational difficulties. Convergence of series solution has been discussed with two illustrated examples, and the method has shown a high-precision, fast approach to solve non-linear (2+1) dimensional PDEs with initial conditions, there is no need any discretization of domain or assumption for a small parameter to be present in the problem. The steps of suggested method are easy implemented, high accuracy and a rapid convergence to the exact solution compared with other methods can be used to solve this type of PDEs.

Strong Solutions for PDE-Based Tomography by Unsupervised Learning

We introduce a novel neural network-based PDEs solver for forward and inverse problems. The solver is grid free, mesh free, and shape free, and the solution is approximated by a neural network. We employ an unsupervised approach such that the input to the network is a point set in an arbitrary domain, and the output is the set of the corresponding function values. The network is trained to minimize deviations of the learned function from the PDE solution and satisfy the boundary conditions. The resulting solution in turn is an explicit, smooth, differentiable function with a known analytical form. We solve the forward problem (observations given the underlying model's parameters), semi-inverse problem (model's parameters given the observations in the whole domain), and full tomography inverse problem (model's parameters given the observations on the boundary) by solving the forward and semi-inverse problems at the same time. The optimized loss function consists of few elements: fidelity term of $L_2$ norm that enforces the PDE in the weak sense, an $L_\infty$ norm term that enforces pointwise fidelity and thus promotes a strong solution, and boundary and initial conditions constraints. It further accommodates regularizers for the solution and/or the model's parameters of the differential operator. This setting is flexible in the sense that regularizers can be tailored to specific problems. We demonstrate our method on several free shape two dimensional (2D) second order systems with application to electrical impedance tomography (EIT) and diffusion equation. Unlike other numerical methods such as finite differences and finite elements, the derivatives of the desired function can be analytically calculated to any order. This framework enables, in principle, the solution of high order and high dimensional nonlinear PDEs.

A multivariate spectral quasi-linearization method for the solution of (2+1) dimensional Burgers’ equations

AbstractIn this paper, we introduce the multi-variate spectral quasi-linearization method which is an extension of the previously reported bivariate spectral quasi-linearization method. The method is a combination of quasi-linearization techniques and the spectral collocation method to solve three-dimensional partial differential equations. We test its applicability on the (2 + 1) dimensional Burgers’ equations. We apply the spectral collocation method to discretize both space variables as well as the time variable. This results in high accuracy in both space and time. Numerical results are compared with known exact solutions as well as results from other papers to confirm the accuracy and efficiency of the method. The results show that the method produces highly accurate solutions and is very efficient for (2 + 1) dimensional PDEs. The efficiency is due to the fact that only few grid points are required to archive high accuracy. The results are portrayed in tables and graphs.

New Approach for Solving Two Dimensional Spaces PDE

In this paper, new approach based on coupled Laplace transformation with decomposition method is proposed to solve type of partial differential equation. Then it’s used to find the accurate solution for heat equation with initial conditions. Four examples introduced to illustrate the accuracy, efficiency of suggested method. The practical results show the importance of suggested method for solve differential equations with high accuracy and easy implemented.

Lump solutions to nonlinear PDEs involving Hirota derivative Dt2DxDy

This paper aims to study lump solutions to a class of (2[Formula: see text]+[Formula: see text]1)-dimensional nonlinear PDE systems, which involve the fourth-order Hirota derivative term: [Formula: see text]. This Hirota derivative term generates higher-order derivatives of the temporal variable. Lump solutions to the resulting new class of nonlinear PDE systems are studied in detail via the Hirota bilinear method.

The relation between the size of perturbations and the dimension of tori in an infinite-dimensional KAM theorem of Pöschel

In this paper, we discuss the relation between the size of perturbations and the dimension of tori based on the KAM theorem of Pöschel (1996) which can be applied to infinite dimensional PDEs, in other words, we give a quantitative version of the KAM theorem of Pöschel (1996). Moreover, motivated by Chung and Yuan (2007, 2008) and Li and Yuan (2012), after infinite KAM iterations, we make action and normal variables of KAM tori of the reduced Hamiltonian system in a domain instead of shrinking them to the origin.

A HJB-POD approach for the control of nonlinear PDEs on a tree structure

The Dynamic Programming approach allows to compute a feedback control for nonlinear problems, but suffers from the curse of dimensionality. The computation of the control relies on the resolution of a nonlinear PDE, the Hamilton-Jacobi-Bellman equation, with the same dimension of the original problem. Recently, a new numerical method to compute the value function on a tree structure has been introduced. The method allows to work without a structured grid and avoids any interpolation.Here, we aim at testing the algorithm for nonlinear two dimensional PDEs. We apply model order reduction to decrease the computational complexity since the tree structure algorithm requires to solve many PDEs. Furthermore, we prove an error estimate which guarantees the convergence of the proposed method. Finally, we show efficiency of the method through numerical tests.

Effect of Viscous Dissipation on MHD Williamson Nanofluid Flow in a Porous Medium

The main objective of this paper is to focus on a numerical study of viscous dissipation effect on the steady state flow of MHD Williamson nanofluid. A mathematical modeled which resembles the physical flow problem has been developed. By using an appropriate transformation, we converted the system of dimensional PDEs (nonlinear) into coupled dimensionless ODEs. The numerical solution of these modeled ordinary differential equations (ODEs) is achieved by utilizing shooting technique together with Adams-Bashforth Moulton method of order four. Finally, the results of discussed for different parameters through graphs and tables.

Moving mesh method for simulating high-dimensional time dependent PDEs with fast propagating shock waves

When solutions of the high dimensional PDE involve fast propagating shock waves, existing variational moving mesh methods are rather unstable and time consuming. To circumvent the problem, we improve the existing variational method by adding one term which adapts the mesh to the dynamic variations of the PDEs during time iterations. This is an extension of the moving mesh strategy in (Gao, 2018) for one dimensional (1-D) problems. We prove the stability and convergence of the present moving mesh equation for arbitrary initial mesh. Consequently, by enlarging the time step sizes of mesh generation, the computational time is shortened without causing instability. The moving mesh equation and the PDE are simulated step by step applying the alternate finite difference method. Both theoretical analysis and numerical results imply that the method is highly effective and robust when simulating high dimensional time dependent PDEs with fast moving shock waves.

Lump and Interaction Solutions to Linear (4+1)-Dimensional PDEs

Taking a class of linear (4+1)-dimensional partial differential equations as examples, we would like to show that there exist lump solutions and interaction solutions in (4+1)-dimensions. We will compute abundant lump solutions and interaction solutions to the considered linear (4+1)-dimensional partial differential equations via symbolic computations, and plot three specific solutions with Maple plot tools, which supplements the existing literature on lump, rogue wave and breather solutions and their interaction solutions in soliton theory.

Lie Symmetry Analysis and Some Exact Solutions of (2+1)-dimensional KdV-Burgers Equation

In this work, the Lie group method is used to perform the similarity reduction and to obtain the exact solutions of the $$(2+1)$$ -dimensional KdV-Burgers equation. Similarity transformation method reduces $$(2+1)$$ -dimensional KdV-Burgers equation into $$(1+1)$$ -dimensional PDEs, later it reduces these PDEs into various ordinary differential equations and helps to find exact solutions of $$(2+1)$$ -dimensional KdV-Burgers equation. With the help of reduced equations, we have obtained the exact explicit solutions. Moreover, later by power series method, the exact analytic solutions of the KdV-Burgers equation are obtained.

Probability measure-valued polynomial diffusions

We introduce a class of probability measure-valued diffusions, coined polynomial, of which the well-known Fleming–Viot process is a particular example. The defining property of finite dimensional polynomial processes considered in [8, 21] is transferred to this infinite dimensional setting. This leads to a representation of conditional marginal moments via a finite dimensional linear PDE, whose spatial dimension corresponds to the degree of the moment. As a result, the tractability of finite dimensional polynomial processes are preserved in this setting. We also obtain a representation of the corresponding extended generators, and prove well-posedness of the associated martingale problems. In particular, uniqueness is obtained from the duality relationship with the PDEs mentioned above.

AN ADAPTIVE MULTIFIDELITY PC-BASED ENSEMBLE KALMAN INVERSION FOR INVERSE PROBLEMS

The ensemble Kalman inversion (EKI), as a derivative-free methodology, has been widely used in the parameter estimation of inverse problems. Unfortunately, its cost may become moderately large for systems described by high dimensional nonlinear PDEs, as EKI requires a relatively large ensemble size to guarantee its performance. In this paper, we propose an adaptive multi-fidelity polynomial chaos (PC) based EKI technique to address this challenge. Our new strategy combines a large number of low-order PC surrogate model evaluations and a small number of high-fidelity forward model evaluations, yielding a multi-fidelity approach. Especially, we present a new approach that adaptively constructs and refines a multi-fidelity PC surrogate during the EKI simulation. Since the forward model evaluations are only required for updating the low-order multi-fidelity PC model, whose number can be much smaller than the total ensemble size of the classic EKI, the entire computational costs are thus significantly reduced. The new algorithm was tested through the two-dimensional time fractional inverse diffusion problems and demonstrated great effectiveness in comparison with PC based EKI and classic EKI.

Lie algebras responsible for zero-curvature representations of scalar evolution equations

Zero-curvature representations (ZCRs) are one of the main tools in the theory of integrable PDEs. In particular, Lax pairs for (1+1)-dimensional PDEs can be interpreted as ZCRs.For any (1+1)-dimensional scalar evolution equation E, we define a family of Lie algebras F(E) which are responsible for all ZCRs of E in the following sense. Representations of the algebras F(E) classify all ZCRs of the equation E up to local gauge transformations. To achieve this, we find a normal form for ZCRs with respect to the action of the group of local gauge transformations.As we show in other publications, using these algebras, one obtains some necessary conditions for integrability of the considered PDEs (where integrability is understood in the sense of soliton theory) and necessary conditions for existence of a Bäcklund transformation between two given equations. Examples of proving non-integrability and applications to obtaining non-existence results for Bäcklund transformations are presented in other publications as well.In our approach, ZCRs may depend on partial derivatives of arbitrary order, which may be higher than the order of the equation E. The algebras F(E) generalize Wahlquist–Estabrook prolongation algebras, which are responsible for a much smaller class of ZCRs.In this paper we describe general properties of F(E) and present generators and relations for these algebras. In other publications we study the structure of F(E) for equations of KdV, Krichever–Novikov, Kaup–Kupershmidt, Sawada–Kotera types. Among the obtained algebras, one finds infinite-dimensional Lie algebras of certain matrix-valued functions on rational and elliptic algebraic curves.

Numerical solutions of two dimensional Sobolev and generalized Benjamin–Bona–Mahony–Burgers equations via Haar wavelets

This work addresses, numerical method based on Haar wavelets and finite differences to solve two dimensional linear, nonlinear Sobolev and non-linear generalized Benjamin–Bona–Mahony–Burgers (NGBBMB) equations. The temporal part is discretized using finite differences while spatial part is approximated by two dimensional Haar wavelets. With this strategy, computing solution of two dimensional PDEs reduces to computing solution of linear system of algebraic equations. Collocation approach is then applied to determine the wavelet coefficients from linear system. This paper shows that two dimensional Haar wavelets are suitable and effective for two dimensional linear and non-linear PDEs. For validation of the proposed scheme different problems have been solved and error norms L∞,L2 are computed. Computation verifies that current scheme has good outcome.

Deep Learning-Based BSDE Solver for Libor Market Model with Applications to Bermudan Swaption Pricing and Hedging

The Libor market model, also known as the BGM Model, is a term structure model of interest rates. It is widely used for pricing interest rate derivatives, especially Bermudan swaptions, and other exotic Libor callable derivatives. For numerical implementation the pricing of derivatives with Libor market models is mainly carried out with Monte Carlo simulation. The PDE grid approach is not particularly feasible due to the “Curse of Dimensionality”. The standard Monte Carlo method for American swaption pricing more or less uses regression to estimate the expected value as a linear combination of basis functions as demonstrated in the classical paper by Longstaff and Schwartz [Longstaff01]. However, [Longstaff01] only provides the lower bound for American option price. Another complexity arises from applying Monte Carlo simulation is the computation of the sensitivities of the option, the so-called “Greeks”, which are fundamental for a trader's hedging activity. Recently, an alternative numerical method based on deep learning and backward stochastic differential equations (BSDEs) appeared in quite a few researches [Weinan17, Jiequn17]. For European style options the feedforward deep neural networks (DNN) show not only feasibility but also efficiency to obtain both prices and numerical Greeks. The standard LMM implementation requires dimension five or higher in factor space even after PCA, which cannot be solved by traditional PDE solvers, such as finite differences or finite elements methods. In this paper, a new backward DNN solver is proposed for Bermudan swaptions. Our approach is representing financial pricing problems in the form of high dimensional stochastic optimal control problems, FBSDEs, or equivalent PDEs. We demonstrate that using backward DNN the high-dimension Bermudan swaption pricing and hedging can be solved effectively and efficiently. A comparison between Monte Carlo simulation and the new method for pricing vanilla interest rate options manifests the superior performance of the new method. We then use this method to calculate prices and Greeks of Bermudan swaptions as a prelude for other Libor callable derivatives.

Local meshless method for convection dominated steady and unsteady partial differential equations

In this paper, we propose a mildly shock-capturing stabilized local meshless method (SLMM) for convection-dominated steady and unsteady PDEs. This work is extension of the numerical procedure, which was designed only for steady state convection-dominated PDEs (Siraj-ul-Islam and Singh in Int J Comput Methods 14(6):1750067, 2017). The proposed meshless methodology is based on employing different type of stencils embodying the already known flow direction. Numerical experiments are performed to compare the proposed method with the finite-difference method on special grid (FDSG) and other numerical methods. Numerical results confirm that the new approach is accurate and efficient for solving a wide class of one- and two- dimensional convection-dominated PDEs having sharp corners and jump discontinuities. Performance of the SLMM is found better in some cases and comparable in other cases, than the mesh-based numerical methods.

Lump and lump-soliton solutions to the Hirota–Satsuma–Ito equation

We apply the Hirota direct method to construct lump and interaction solutions to the Hirota–Satsuma–Ito (HSI) equation. We establish a general theory for finding the lump-soliton to (2+1)-dimensional nonlinear PDEs. We generate the corresponding lump and lump-soliton solutions to the HSI equation by the logarithm transformation of the dependent variables.

Abundant lumps and their interaction solutions of (3+1)-dimensional linear PDEs

The paper aims to explore the existence of diverse lump and interaction solutions to linear partial differential equations in (3+1)-dimensions. The remarkable richness of exact solutions to a class of linear partial differential equations in (3+1)-dimensions will be exhibited through Maple symbolic computations, which yields exact lump, lump-periodic and lump–soliton solutions. The results expand the understanding of lump, freak wave and breather solutions and their interaction solutions in soliton theory.