We consider solving the forward and inverse partial differential equations(PDEs)which have sharp solutions with physics-informed neural networks(PINNs)in this *** particular,to better capture the sharpness of the solu...
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We consider solving the forward and inverse partial differential equations(PDEs)which have sharp solutions with physics-informed neural networks(PINNs)in this *** particular,to better capture the sharpness of the solution,we propose the adaptive sampling methods(ASMs)based on the residual and the gradient of the *** first present a residual only-based ASM denoted by ASMⅠ.In this approach,we first train the neural network using a small number of residual points and divide the computational domain into a certain number of sub-domains,then we add new residual points in the sub-domain which has the largest mean absolute value of the residual,and those points which have the largest absolute values of the residual in this sub-domain as new residual *** further develop a second type of ASM(denoted by ASMⅡ)based on both the residual and the gradient of the solution due to the fact that only the residual may not be able to efficiently capture the sharpness of the *** procedure of ASMⅡis almost the same as that of ASMⅠ,and we add new residual points which have not only large residuals but also large *** demonstrate the effectiveness of the present methods,we use both ASMⅠand ASMⅡto solve a number of PDEs,including the Burger equation,the compressible Euler equation,the Poisson equation over an Lshape domain as well as the high-dimensional Poisson *** has been shown from the numerical results that the sharp solutions can be well approximated by using either ASMⅠor ASMⅡ,and both methods deliver much more accurate solutions than the original PINNs with the same number of residual ***,the ASMⅡalgorithm has better performance in terms of accuracy,efficiency,and stability compared with the ASMⅠ*** means that the gradient of the solution improves the stability and efficiency of the adaptive sampling procedure as well as the accuracy of the ***,we also employ the similar adaptive sampling technique fo
In this article, we study two proposed fractional models. Initially, we introduce a new generalized Marshall–Hoare model using the Mittag-Leffler type function to simulate the cooling of the internal solid organs dur...
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The main objective of this research is to discuss the existence of solutions for a nonlocal hybrid boundary value problem of ψ -Caputo fractional differential equations. To prove this result, we use Darbo’s fixed po...
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In this paper we obtain the boundedness of non-regular pseudo-differential operators with symbols in Besov spaces on matrix-weighted Besov-Triebel-Lizorkin *** symbols include the classical Hörmander classes.
In this paper we obtain the boundedness of non-regular pseudo-differential operators with symbols in Besov spaces on matrix-weighted Besov-Triebel-Lizorkin *** symbols include the classical Hörmander classes.
The purpose of this work is to establishes the existence and uniqueness of the Schrödinger problem solution in the extended Colombeau algebra Ge. Then we look at the association notion in conjunction with the cla...
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In this work, we have explored a fractional Newton’s Second Law of motion involving the ψ-Caputo operator of order α∈(1,2]. We proved the existence and uniqueness of solutions for different classes of force functi...
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This paper mainly studies the contact extension of conservative or dissipative systems, including some old and new results for wholeness. Then extension of contact system is corresponding to the symplectification of c...
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This paper mainly studies the contact extension of conservative or dissipative systems, including some old and new results for wholeness. Then extension of contact system is corresponding to the symplectification of contact Hamiltonian system. This is a reciprocal process and the relation between symplectic system and contact system has been discussed. We have an interesting discovery that by adding a pure variable p,the slope of the tangent of the orbit, every differential system can be regarded as an independent subsystem of contact Hamiltonian system defined on the projection space of contact phase space.
In this paper,we construct a new cell-centered nonlinear finite volume scheme that preserves the extremum principle for heterogeneous anisotropic diffusion equation on distorted *** introduce a new nonlinear approach ...
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In this paper,we construct a new cell-centered nonlinear finite volume scheme that preserves the extremum principle for heterogeneous anisotropic diffusion equation on distorted *** introduce a new nonlinear approach to construct the conservative flux,that is,a linear second order flux is firstly given and a nonlinear conservative flux is then constructed by using an adaptive method and a nonlinear weighted *** new scheme does not need to use the convex combination of the cell-center unknowns to approximate the auxiliary unknowns,so it can deal with the problem with general discontinuous *** results show that our new scheme performs more robust than some existing schemes on highly distorted meshes.
In this paper we first propose a phase-field model for the containerless freezing problems, in which the volume expansion or shrinkage of the liquid caused by the density change during the phase change process is cons...
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In this paper we first propose a phase-field model for the containerless freezing problems, in which the volume expansion or shrinkage of the liquid caused by the density change during the phase change process is considered by adding a mass source term to the continuum equation. Then a phase-field-based lattice Boltzmann (LB) method is further developed to simulate solid-liquid phase change phenomena in multiphase systems. We test the developed LB method by the problem of conduction-induced freezing in a semi-infinite space, the three-phase Stefan problem, and the droplet solidification on a cold surface, and the numerical results are in agreement with the analytical and experimental solutions. In addition, the LB method is also used to study the rising bubbles with solidification. The results of the present method not only accurately capture the effect of bubbles on the solidification process, but also are in agreement with the previous work. Finally, a parametric study is carried out to examine the influences of some physical parameters on the sessile droplet solidification, and it is found that the time of droplet solidification increases with the increase of droplet volume and contact angle.
We prove the positive mass theorem for asymptotically flat(AF for short) manifolds with finitely many isolated conical singularities. We do not impose the spin condition. Instead, we use the conformal blow-up techniqu...
We prove the positive mass theorem for asymptotically flat(AF for short) manifolds with finitely many isolated conical singularities. We do not impose the spin condition. Instead, we use the conformal blow-up technique which dates back to Schoen's final resolution of the Yamabe conjecture.
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