Open Access. Powered by Scholars. Published by Universities.®

Numerical Analysis and Computation Commons™

Open Access. Powered by Scholars. Published by Universities.®

Partial differential equations

Articles 1 - 8 of 8

Full-Text Articles in Numerical Analysis and Computation

Stochastic Derivative-Free Deep Learning Methods For Solving High Dimensional Partial Differential Equations, Qing He Mr. May 2026

Stochastic Derivative-Free Deep Learning Methods For Solving High Dimensional Partial Differential Equations, Qing He Mr.

Mathematics Theses and Dissertations

Solving high-dimensional partial differential equations (PDEs) is a fundamental challenge in scientific computing, with applications ranging from quantum chemistry and computational finance to statistical physics and stochastic optimal control.  Classical numerical methods such as finite element or finite difference schemes suffer from the curse of dimensionality, rendering them computationally infeasible when the dimension $d$ exceeds a handful. Physics-informed neural network (PINN) methods alleviate this by embedding the PDE residual directly into a loss function, but they require computing derivatives of the network with respect to its spatial inputs---an operation that scales poorly in high dimensions and demands that the approximate …

Go to article

Solution Of Fractional Order Diffusion Equations With Clique Neural Network, Merve Zeynep Kaya, Mesut Karabacak, Ercan Çelik Sep 2025

Solution Of Fractional Order Diffusion Equations With Clique Neural Network, Merve Zeynep Kaya, Mesut Karabacak, Ercan Çelik

Mathematical Modelling and Numerical Simulation with Applications

In this paper, the clique artificial neural network method is used to solve the fractional diffusion equation, which is a subclass of partial differential equations. The clique neural network architecture is constructed using input, hidden, and output layers. Several degrees of clique polynomials were used as activation functions, and the output layer was obtained by multiplying them with weight coefficients. Subsequently, the optimization equation was derived, and the exact solution, numerical solution, and error function graphs were obtained using a specialized algorithm. Analysis of the results demonstrates that the clique artificial neural network method provides quicker and more accurate results …

Go to article

(R2067) Solutions Of Hyperbolic System Of Time Fractional Partial Differential Equations For Heat Propagation, Sagar Sankeshwari, Vinayak Kulkarni Jun 2024

(R2067) Solutions Of Hyperbolic System Of Time Fractional Partial Differential Equations For Heat Propagation, Sagar Sankeshwari, Vinayak Kulkarni

Applications and Applied Mathematics: An International Journal (AAM)

Hyperbolic linear theory of heat propagation has been established in the framework of a Caputo time fractional order derivative. The solution of a system of integer and fractional order initial value problems is achieved by employing the Adomian decomposition approach. The obtained solution is in convergent infinite series form, demonstrating the method’s strengths in solving fractional differential equations. Moreover, the double Laplace transform method is employed to acquire the solution of a system of integer and fractional order boundary conditions in the Laplace domain. An inversion of double Laplace transforms has been achieved numerically by employing the Xiao algorithm in …

Go to article

Modified Gaussian Radial Basis Function Method For The Burgers Systems, Hossein Aminikhah, Mostafa Sadeghi Jun 2019

Modified Gaussian Radial Basis Function Method For The Burgers Systems, Hossein Aminikhah, Mostafa Sadeghi

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, the systems of variable-coefficient coupled Burgers equation are solved by a free mesh method. The method is based on the collocation points with the modified Gaussian (MGA) radial basis function (RBF). Dependent parameters and independent parameters and their effect on the stability are shown. The accuracy and efficiency of the method has been checked by two examples. The results of numerical experiments are compared with analytical solutions by calculating errors infinity-norm.

Go to article

Algorithms To Approximate Solutions Of Poisson's Equation In Three Dimensions, Ray Dambrose Mar 2019

Algorithms To Approximate Solutions Of Poisson's Equation In Three Dimensions, Ray Dambrose

Rose-Hulman Undergraduate Mathematics Journal

The focus of this research was to develop numerical algorithms to approximate solutions of Poisson's equation in three dimensional rectangular prism domains. Numerical analysis of partial differential equations is vital to understanding and modeling these complex problems. Poisson's equation can be approximated with a finite difference approximation. A system of equations can be formed that gives solutions at internal points of the domain. A computer program was developed to solve this system with inputs such as boundary conditions and a nonhomogenous source function. Approximate solutions are compared with exact solutions to prove their accuracy. The program is tested with an …

Go to article

On The Well-Posedness And Global Boundary Controllability Of A Nonlinear Beam Model, Jessie Jamieson Aug 2018

On The Well-Posedness And Global Boundary Controllability Of A Nonlinear Beam Model, Jessie Jamieson

Department of Mathematics: Dissertations, Theses, and Student Research

The theory of beams and plates has been long established due to works spanning many fields, and has been explored through many investigations of beam and plate mechanics, controls, stability, and the well-posedness of systems of equations governing the motions of plates and beams. Additionally, recent investigations of flutter phenomena by Dowell, Webster et al. have reignited interest into the mechanics and stability of nonlinear beams. In this thesis, we wish to revisit the seminal well-posedness results of Lagnese and Leugering for the one dimensional, nonlinear beam from their 1991 paper, "Uniform stabilization of a nonlinear beam by nonlinear boundary …

Go to article

A Numerical Study Of Construction Of Honey Bee Comb, Pamela Guerrero, Pamela C. Guerrero Jan 2017

A Numerical Study Of Construction Of Honey Bee Comb, Pamela Guerrero, Pamela C. Guerrero

Murray State Theses and Dissertations

We use finite difference methods in the treatment of an existing system of partial differential equations that captures the dynamics of parallel honeycomb construction in a bee hive. We conduct an uncertainty analysis by calculating the partial rank correlation coefficient for the parameters to find which are most important to the outcomes of the model. We then use an eFAST method to determine both the individual and total sensitivity index for the parameters. Afterwards we examine our numerical model under varying initial conditions and parameter values, and compare ratios found from local data with the golden mean by fitting images …

Go to article

A Local Radial Basis Function Method For The Numerical Solution Of Partial Differential Equations, Maggie Elizabeth Chenoweth Jan 2012

A Local Radial Basis Function Method For The Numerical Solution Of Partial Differential Equations, Maggie Elizabeth Chenoweth

Theses, Dissertations and Capstones

Most traditional numerical methods for approximating the solutions of problems in science, engineering, and mathematics require the data to be arranged in a structured pattern and to be contained in a simply shaped region, such as a rectangle or circle. In many important applications, this severe restriction on structure cannot be met, and traditional numerical methods cannot be applied. In the 1970s, radial basis function (RBF) methods were developed to overcome the structure requirements of existing numerical methods. RBF methods are applicable with scattered data locations. As a result, the shape of the domain may be determined by the application …

Go to article