Exact Solution of Navier-Stokes Equations

In Navier-Stokes equations (NASA’s Navier-Stokes Equations, 3-dimensional-unsteady), wed iscover the exact solution by Newton potential function and time-function. We think the solution likely Newton potential function that be able to solve Laplace equation

Analytical Solution of the Three-Dimensional Laplace Equation in Terms of Linear Combinations of Hypergeometric Functions

We present some solutions of the three-dimensional Laplace equation in terms of linear combinations of generalized hyperogeometric functions in prolate elliptic geometry, which simulates the current tokamak shapes. Such solutions are valid for particular parameter values. The derived solutions are compared with the solutions obtained in the standard toroidal geometry.

GPU-Accelerated Laplace Equation Model Development Based on CUDA Fortran

In this study, a CUDA Fortran-based GPU-accelerated Laplace equation model was developed and applied to several cases. The Laplace equation is one of the equations that can physically analyze the groundwater flows, and is an equation that can provide analytical solutions. Such a numerical model requires a large amount of data to physically regenerate the flow with high accuracy, and requires computational time. These numerical models require a large amount of data to physically reproduce the flow with high accuracy and require computational time. As a way to shorten the computation time by applying CUDA technology, large-scale parallel computations were performed on the GPU, and a program was written to reduce the number of data transfers between the CPU and GPU. A GPU consists of many ALUs specialized in graphic processing, and can perform more concurrent computations than a CPU using multiple ALUs. The computation results of the GPU-accelerated model were compared with the analytical solution of the Laplace equation to verify the accuracy. The computation results of the GPU-accelerated Laplace equation model were in good agreement with the analytical solution. As the number of grids increased, the computational time of the GPU-accelerated model gradually reduced compared to the computational time of the CPU-based Laplace equation model. As a result, the computational time of the GPU-accelerated Laplace equation model was reduced by up to about 50 times.

A strong maximum principle for the fractional laplace equation with mixed boundary condition

In this work we prove a strong maximum principle for fractional elliptic problems with mixed Dirichlet–Neumann boundary data which extends the one proved by J. Dávila (cf. [11]) to the fractional setting. In particular, we present a comparison result for two solutions of the fractional Laplace equation involving the spectral fractional Laplacian endowed with homogeneous mixed boundary condition. This result represents a non–local counterpart to a Hopf’s Lemma for fractional elliptic problems with mixed boundary data.

Numerical Results of Crank-Nicolson and Implicit Schemes to Laplace Equation with Uniform and Non-Uniform Grids

In this paper, we investigate the numerical results between Implicit and Crank-Nicolson method for Laplace equation. Based on the numerical results obtained, we get the conclusion that the absolute error of Crank-Nicolson method is smaller than the absolute error of Implicit method for uniform and non-uniform grids which both refer to the analytical solution of Laplace equation obtained by separable variable method.Keywords: Crank-Nicolson; Implicit; Laplace equation; separable variable method; uniform and non-uniform grids. AbstrakDalam makalah ini, kami menyelidiki hasil numerik antara etode Implisit dan Crank-Nicolson untuk persamaan Laplace. Berdasarkan hasil numerik yang diperoleh, kita mendapatkan kesimpulan bahwa kesalahan absolut metode Crank-Nicolson lebih kecil daripada kesalahan absolut metode Implisit untuk grid seragam dan tak-seragam yang keduanya mengacu pada solusi analitik persamaan Laplace yang diperoleh dengan metode separable.Kata kunci: Crank-Nicolson; Implisit; persamaan Laplace; metode variable terpisah; grid seragam dan tak-seragam.