scholarly journals Solutions of Smooth Nonlinear Partial Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-37
Author(s):  
Jan Harm van der Walt
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Spatial Dimension ◽  
Nonlinear Pdes ◽  
Generalized Solutions ◽  
Structure Of Solutions ◽  
Partial Differential ◽  
Convergence Spaces ◽  
Order Completion

The method of order completion provides a general and type-independent theory for the existence and basic regularity of the solutions of large classes of systems of nonlinear partial differential equations (PDEs). Recently, the application of convergence spaces to this theory resulted in a significant improvement upon the regularity of the solutions and provided new insight into the structure of solutions. In this paper, we show how this method may be adapted so as to allow for the infinite differentiability of generalized functions. Moreover, it is shown that a large class of smooth nonlinear PDEs admit generalized solutions in the space constructed here. As an indication of how the general theory can be applied to particular nonlinear equations, we construct generalized solutions of the parametrically driven, damped nonlinear Schrödinger equation in one spatial dimension.

scholarly journals Optimal Combination of EEFs for the Model Reduction of Nonlinear Partial Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Jun Shuai ◽  
Xuli Han
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Model Reduction ◽  
Nonlinear Partial Differential Equations ◽  
Optimization Techniques ◽  
Optimal Combination ◽  
Nonlinear Pdes ◽  
Partial Differential ◽  
Tiny Amount ◽  
Low Dimensional

Proper orthogonal decomposition is a popular approach for determining the principal spatial structures from the measured data. Generally, model reduction using empirical eigenfunctions (EEFs) can generate a relatively low-dimensional model among all linear expansions. However, the neglectful modes representing only a tiny amount of energy will be crucial in the modeling for certain type of nonlinear partial differential equations (PDEs). In this paper, an optimal combination of EEFs is proposed for model reduction of nonlinear partial differential equations (PDEs), obtained by the basis function transformation from the initial EEFs. The transformation matrix is derived from straightforward optimization techniques. The present new EEFs can keep the dynamical information of neglectful modes and generate a lower-dimensional and more precise dynamical system for the PDEs. The numerical example shows its effectiveness and feasibility for model reduction of the nonlinear PDEs.

Extended tanh-Function Method for Finding Travelling Wave Solutions of Some Nonlinear Partial Differential Equations

2005 ◽  
Vol 60 (1-2) ◽  
pp. 7-16 ◽  
Author(s):  
Mustafa Inc ◽  
Engui G. Fan
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Function Method ◽  
Nonlinear Partial Differential Equations ◽  
Travelling Wave ◽  
Nonlinear Pdes ◽  
Travelling Wave Solutions ◽  
Partial Differential ◽  
Wave Solutions

In this paper, we find travelling wave solutions of some nonlinear partial differential equations (PDEs) by using the extended tanh-function method. Some illustrative equations are investigated by this method and new travelling wave solutions are found. In addition, the properties of these nonlinear PDEs are shown with some figures.

scholarly journals Bayesian numerical methods for nonlinear partial differential equations

2021 ◽  
Vol 31 (5) ◽  
Author(s):  
Junyang Wang ◽  
Jon Cockayne ◽  
Oksana Chkrebtii ◽  
T. J. Sullivan ◽  
Chris. J. Oates
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Nonlinear Partial Differential Equations ◽  
Nonlinear Pdes ◽  
Partial Differential ◽  
Inference Problem ◽  
Probabilistic Uncertainty ◽  
Right Hand ◽  
The Right

AbstractThe numerical solution of differential equations can be formulated as an inference problem to which formal statistical approaches can be applied. However, nonlinear partial differential equations (PDEs) pose substantial challenges from an inferential perspective, most notably the absence of explicit conditioning formula. This paper extends earlier work on linear PDEs to a general class of initial value problems specified by nonlinear PDEs, motivated by problems for which evaluations of the right-hand-side, initial conditions, or boundary conditions of the PDE have a high computational cost. The proposed method can be viewed as exact Bayesian inference under an approximate likelihood, which is based on discretisation of the nonlinear differential operator. Proof-of-concept experimental results demonstrate that meaningful probabilistic uncertainty quantification for the unknown solution of the PDE can be performed, while controlling the number of times the right-hand-side, initial and boundary conditions are evaluated. A suitable prior model for the solution of PDEs is identified using novel theoretical analysis of the sample path properties of Matérn processes, which may be of independent interest.

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