High-dimensional Equations Research Articles

The High Dimensional Fisher-KPP Nonlocal Diffusion Equation with Free Boundary and Radial Symmetry, Part 1

This is Part 1 of a two-part series, where we study the radially symmetric high dimensional Fisher-KPP nonlocal diffusion equation with free boundary, and obtain a rather complete description of its long-time dynamical behavior, which reveals both similarities and fundamental differences to its one dimensional version considered in [J. Cao, Y. Du, F. Li, and W. T. Li, J. Funct. Anal., 277 (2019), pp. 2772--2814; Y. Du, F. Li, and M. Zhou, J. Math. Pure Appl., 154 (2021), pp. 30--66; Y. Du and W. Ni, Rate of Propagation for the Fisher-KPP Equation with Nonlocal Diffusion, and Free Boundaries, preprint, 2021] recently. Our goals on the long-time dynamics of the model include (a) find the threshold condition on the kernel function that governs the onset of accelerated spreading, (b) determine the spreading speed when it is finite, (c) obtain sharp estimates of the spreading profile for finite speed spreading as well as for accelerated spreading. The tasks in (a) and (b) are carried out here in Part 1, while (c) is done in a separate Part 2. This high dimensional problem poses considerable technical difficulties when the rate of spreading is considered, and we overcome that by introducing an intermediate kernel function which plays a crucial role both in determining the onset condition for accelerated spreading and in obtaining the spreading speed when it is finite.

The Defocusing Energy-Supercritical Nonlinear Schrödinger Equation in High Dimensions

The Defocusing Energy-Supercritical Nonlinear Schrödinger Equation in High Dimensions

On the Analytical Treatment for the Fractional-Order Coupled Partial Differential Equations via Fixed Point Formulation and Generalized Fractional Derivative Operators

High-dimensional fractional equation investigation is a cutting-edge discipline with considerable pragmatic and speculative consequences in engineering, epidemiology, and other scientific disciplines. In this study, a hybrid Jafari transform mixed with the Adomian decomposition method for obtaining the analytical solution to Burgers’ problem is provided. Burgers’ equation is a vital mathematical expression that appears in a variety of computational modelling fields, including fluid mechanics, nonlinear acoustics, gas dynamics, and traffic flow. By considering a hybrid transform, semianalytical techniques are constructed for the Caputo and Atangana-Baleanu fractional derivative operators. Besides that, existence and uniqueness analyses are carried out with the aid of the Banach contraction-fixed point theory. To obtain the models’ findings, we employed the Jafari transform on fractional-order Burger equations (BEs), supplemented by the inverse Jafari transform. The projected findings for the fractional BEs have been depicted visually. Ultimately, numerical figures are provided to validate the practicality and efficacy. The solution obtained by employing the supplied methodologies has been validated to have the appropriate rate of convergence to the precise solution. The main advantage of the suggested method is the relatively small number of computations performed. It can also be used to address fractional-order scientific issues in a multitude of fields.

Dynamics of soliton waves of group-invariant solutions through optimal system of an extended KP-like equation in higher dimensions with applications in medical sciences and mathematical physics

The dynamics of solitons waves associated with higher-order nonlinear partial differential equations that model situations in physical systems obtainable in the fields of science and engineering help to understand the physical meaning of various soliton solutions obtained for these nonlinear differential equations. Thus, in this paper, we analytically investigate an extended Kadomtsev-Petviashvili-like equation existent in three dimensions. The robust technique of the Lie group theory of differential equation was invoked to achieve analytic solutions to the equation. This technique is used in a systematic way to generate the Lie point symmetries of the equation under study. Consequently, an optimal system of Lie subalgebra related to the equation is obtained. Thereafter, we engage the various gained subalgebras to reduce the Kadomtsev-Petviashvili-like equation to ordinary differential equations for the possibility of obtaining relevant closed-form solutions. Fortunately, various soliton solutions were found. These include different complex soliton solutions consisting of dark, bright, topological kink and singular. Some other solutions achieved are logarithmic, periodic and those which contain arbitrary functions. Therefore, to understand the physical meaning of these solutions, we depict them graphically. This exposed us to various wave structures which were later analyzed and applied. Moreover, we highlighted the significance of these solutions in various fields of science and engineering.

New Solutions of Nonlinear Dispersive Equation in Higher-Dimensional Space with Three Types of Local Derivatives

The aim of this paper is to use the Nucci’s reduction method to obtain some novel exact solutions to the s-dimensional generalized nonlinear dispersive mK(m,n) equation. To the best of the authors’ knowledge, this paper is the first work on the study of differential equations with local derivatives using the reduction technique. This higher-dimensional equation is considered with three types of local derivatives in the temporal sense. Different types of exact solutions in five cases are reported. Furthermore, with the help of the Maple package, the solutions found in this study are verified. Finally, several interesting 3D, 2D and density plots are demonstrated to visualize the nonlinear wave structures more efficiently.

Multiple nodal solutions for a class of Kirchhoff-type equations in high dimensions

Multiple nodal solutions for a class of Kirchhoff-type equations in high dimensions

A group theoretic analysis on heat transfer in MHD thermally slip Carreau fluid subject to multiple flow regimes (MFRs)

It is well believed that computational modelling is the best remedy to study heat transfer in complicated manmade or naturally occurring phenomena claimed in general sciences. Therefore, owning such an important belief, we proposed mathematical modelling and corresponding solution of heat transfer in Carreau fluid flow with physical effects namely, heat absorption and generation, thermal slip, magnetic field, porous medium, velocity slip, and chemical reaction. The Carreau flow field is controlled by constructing mathematical equations. Lie symmetry is used to obtain the particular scaling group of transformations. These transformations are carried to step down the higher dimensional equations in terms of ordinary differential equations. Carreau fluid velocity, Carreau temperature, Carreau concentration, Nusselt number, Sherwood number, and skin friction dependency are examined towards involved flow variables. The surface physical quantities are investigated in various frames namely, porous medium, non-porous medium, magnetized flow field, non-magnetized flow field, thermal slip, and non-thermal slip fields. For Carreau fluid flow field we have noticed that the Nusselt number shows direct relation with Prandtl number variation in both non-porous and porous mediums.

Normalized solutions for nonlinear Kirchhoff type equations in high dimensions

<abstract><p>We study the normalized solutions for nonlinear Kirchhoff equation with Sobolev critical exponent in high dimensions $ \mathbb{R}^N(N\geqslant4) $. In particular, in dimension $ N = 4 $, there is a special phenomenon for Kirchhoff equation that the mass critical exponent $ 2+\frac{8}{N} $ is equal to the energy critical exponent $ \frac{2N}{N-2} $, which leads to the fact that the equation no longer has a variational structure in dimensions $ N\geqslant 4 $ if we consider the mass supercritical case, and remains unsolved in the existing literature. In this paper, by using appropriate transform, we first get the equivalent system of Kirchhoff equation. With the equivalence result, we obtain the nonexistence, existence and multiplicity of normalized solutions by variational methods, Cardano's formulas and Pohožaev identity.</p></abstract>

Two Physics-Based Schwarz Preconditioners for Three-Temperature Radiation Diffusion Equations in High Dimensions

Two Physics-Based Schwarz Preconditioners for Three-Temperature Radiation Diffusion Equations in High Dimensions

Determination of monopile offshore structure response to stochastic wave loads via analog filter approximation and GV-GDEE procedure

Efficient and accurate stochastic response analysis of offshore structures subjected to ocean wave loads has been a challenging problem for several decades. This is especially true for multi-degree-of-freedom nonlinear structures when information about the probability density is desired. How to account for the random field of the wave loads, and how to deal with the coupling of high-dimensional partial differential equation governing the joint probability density functions are among the major challenges. To this end, in the present paper an approach is proposed by incorporating the filter approximation of random fields of sea waves with the recently developed Globally-evolving based generalized density evolution equation (GV-GDEE). To deal with the first issue regarding the random wave filed, the analog filter technique is adopted. In particular, a filter approximation of the wave kinematics field distributed over a vertical line along the ocean domain is proposed. It comprises a linear differential equation set with white noise input. An augmented system is thereby constructed by incorporating the additional differential equation set yielding wave excitation simulated by an analog filter into the equation of motion of a monopile offshore structure. To circumvent the issue of coupling of high-dimensional equations for this augmented system with even higher dimension, the GV-GDEE approach is employed. Specifically, by constructing the effective drift coefficients based on the high-dimensional drift coefficients in the associated Fokker-Planck-Kolmogorov (FPK) equation, a two-dimensional GV-GDEE in terms of the response quantity of interest is derived. Further, this GV-GDEE is solved by the path integral method. Furthermore, two numerical examples are included. In particular, the nonstationary stochastic responses of a 5 megawatt (MW) fixed monopile offshore wind turbine structure under two operating conditions are studied. Comparisons with pertinent Monte Carlo simulations (MCS) demonstrate the accuracy and efficiency of the proposed approach.

NUMERICAL SIMULATION OF THE MOVEMENT OF MOISTURE IN THE SOIL

By constructing the method of collocation of radial basis functions in combination with the method of differences, a two-dimensional mathematical model with boundary conditions for the movement of soil water during irrigation is proposed. The nonlinear term is considered by the difference method, and the equation is solved using an implicit scheme. In addition, the existence and uniqueness of the solution of the equation of motion of groundwater is proved. Radial basis functions, also known as distance basis functions, are a type of functions with a base variable. They are isotropic and simple in form and can be easily solved by numerical calculation. The method combining radial basis functions with collocation has many advantages in solving partial differential equations. Numerical results show that the proposed method has high accuracy and is easier to use than traditional methods. In addition, the choice of parameter c plays an important role in ensuring the accuracy of calculations. This lays the foundation for numerical solutions of high-dimensional equations of groundwater motion.

Numerical solution for three-dimensional nonlinear mixed Volterra–Fredholm integral equations via modified moving least-square method

The major purpose of this article is to expand the three-dimensional (3D) modified moving least-square method for the numerical solution of 3D linear and nonlinear Volterra–Fredholm integral equations of the second kind. This approach is very convenient for solving integral equations in high dimensions and it does not require any need for mesh connectivity. The size of the support used is the only factor that has a significant effect on the maximum errors of the MLS method. To overcome this problem, the MMLS approach with a non-singular moment matrix is applied to obtain better results than MLS approximation on using the best support, then applying the method in three dimensions can be easily achieved. The numerical experiments of the MMLS and classical MLS techniques are presented to show the difference between both methods for multi-dimensional problems. The convergence analysis is provided and some numerical tests are given to prove the applicability of this technique.

On singular Abreu equations in higher dimensions

We study the solvability of the second boundary value problem of a class of highly singular, fully nonlinear fourth order equations of Abreu type in higher dimensions under either a smallness condition or radial symmetry.

A new randomized vector algorithm for iterative solution of large linear systems

In this letter we suggest a new randomized scalable stochastic-matrix-based algorithms for calculation of large matrix iterations. Special focus is on positive or irreducible nonnegative class of matrices. As an application, a new randomized vector algorithm for iterative solution of large linear systems of algebraic equations governed by M-matrices is constructed. The idea behind these stochastic methods is in a randomized vector representation of matrix iterations. The iterations are performed by sampling random columns only, thus avoiding not only matrix but also matrix vector multiplications. As a result, the algorithm is highly efficient for solving linear equations of high dimension, its computational cost depends linearly on the dimension. Extensions of the suggested randomized iteration method to general classes of matrices are also discussed.

Solving a Higher-Dimensional Time-Fractional Diffusion Equation via the Fractional Reduced Differential Transform Method

In this study, exact and approximate solutions of higher-dimensional time-fractional diffusion equations were obtained using a relatively new method, the fractional reduced differential transform method (FRDTM). The exact solutions can be found with the benefit of a special function, and we applied Caputo fractional derivatives in this method. The numerical results and graphical representations specified that the proposed method is very effective for solving fractional diffusion equations in higher dimensions.

On the nature of Hawking’s incompleteness for the Einstein-vacuum equations: The regime of moderately spatially anisotropic initial data

In the mathematical physics literature, there are heuristic arguments, going back three decades, suggesting that for an open set of initially smooth solutions to the Einstein-vacuum equations in high dimensions, stable, approximately monotonic curvature singularities can dynamically form along a spacelike hypersurface. In this article, we study the Cauchy problem and give a rigorous proof of this phenomenon in sufficiently high dimensions, thereby providing the first constructive proof of stable curvature-blowup (without symmetry assumptions) along a spacelike hypersurface as an effect of pure gravity. Our proof applies to an open subset of regular initial data satisfying the assumptions of Hawking’s celebrated “singularity” theorem, which shows that the solution is geodesically incomplete but does not reveal the nature of the incompleteness. Specifically, our main result is a proof of the dynamic stability of the Kasner curvature singularity for a subset of Kasner solutions whose metrics exhibit only moderately (as opposed to severely) spatially anisotropic behavior. Of independent interest is our method of proof, which is more robust than earlier approaches in that (i) it does not rely on approximate monotonicity identities and (ii) it accommodates the possibility that the solution develops very singular high-order spatial derivatives, whose blowup-rates are allowed to be, within the scope of our bootstrap argument, much worse than those of the base-level quantities driving the fundamental blowup. For these reasons, our approach could be used to obtain similar blowup-results for various Einstein-matter systems in any number of spatial dimensions for solutions corresponding to an open set of moderately spatially anisotropic initial data, thus going beyond the nearly spatially isotropic regime treated in earlier works.

The blow-up rate for a non-scaling invariant semilinear wave equations in higher dimensions

We consider the semilinear wave equation ∂t2u−Δu=f(u),(x,t)∈RN×[0,T),(1) with f(u)=|u|p−1uloga(2+u2), where p>1 and a∈R, with subconformal power nonlinearity. We will show that the blow-up rate of any singular solution of (1) is given by the ODE solution associated with (1), The result in one space dimension, has been proved in Hamza and Zaag (2020). Our goal here is to extend this result to higher dimensions.

An extension of optimal auxiliary function method to fractional order high dimensional equations

In this article, the new approach called, the optimal auxiliary function method (OAFM) has been extended with the help of the fractional complex transform method (FCTM) to fractional order partial differential equations (PDEs). The beauty of the method is that, there is no need for small or large parameter assumption in the problem. Similarly, the proposed method gives an approximate solution after only one itearion. To test the proposed method, we take the time-fractional order Zakharov-Kuznetsov equations as a test example. The computed results of the suggested method are compared with the new iterative method (NIM), perturbation iteration method (PIA), and Residual power series method (RPSM). The accuracy and effectiveness of the proposed method can be proven by numerical and graphical results.

An adaptive sparse grid local discontinuous Galerkin method for Hamilton-Jacobi equations in high dimensions

The Hamilton-Jacobi (HJ) equations arise in optimal control and many other applications. Oftentimes, such equations are posed in high dimensions, and this presents great numerical challenges. In this paper, we propose an adaptive sparse grid (also called adaptive multiresolution) local discontinuous Galerkin (DG) method for solving Hamilton-Jacobi equations in high dimensions. By using the sparse grid techniques, we can treat moderately high dimensional cases. Adaptivity is incorporated to capture kinks and other local structures of the solutions. Two classes of multiwavelets including the orthonormal Alpert's multiwavelets and the interpolatory multiwavelets are used to achieve multiresolution. Numerical tests in up to four dimensions are provided to validate the performance of the method.

Finite volume approximation with ADI scheme and low-rank solver for high dimensional spatial distributed-order fractional diffusion equations

High dimensional conservative spatial distributed-order fractional diffusion equation is discretized by midpoint quadrature rule, Crank–Nicolson method, and a finite volume approximation, with alternating direction implicit scheme. The resulting scheme is shown to be consistent and unconditionally stable, hence convergent with order 3−α, where α is the maximum of the involving fractional orders. Moreover, if the initial condition and source term possess Tensor-Train format (TT-format) with low TT-ranks, the scheme can be solved in TT-format, such that higher dimensional cases can be considered. Perturbation analysis ensures that the accumulated errors due to data recompression do not affect the overall convergence order. Numerical examples with low TT-ranks initial conditions and source terms, and with dimensions up to 20 are tested.