scholarly journals Solving PDEs with Intrepid

2012 ◽  
Vol 20 (2) ◽  
pp. 151-180 ◽  
Author(s):  
P. Bochev ◽  
H.C. Edwards ◽  
R.C. Kirby ◽  
K. Peterson ◽  
D. Ridzal
Keyword(s):  
Data Analysis ◽  
Partial Differential Equations ◽  
Numerical Methods ◽  
Differential Equations ◽  
Software Design ◽  
General Context ◽  
Partial Differential ◽  
Mathematical Ideas ◽  
Wide Range ◽  
Local Cell

Intrepid is a Trilinos package for advanced discretizations of Partial Differential Equations (PDEs). The package provides a comprehensive set of tools for local, cell-based construction of a wide range of numerical methods for PDEs. This paper describes the mathematical ideas and software design principles incorporated in the package. We also provide representative examples showcasing the use of Intrepid both in the context of numerical PDEs and the more general context of data analysis.

A method for the numerical or mechanical solution of certain types of partial differential equations

1937 ◽  
Vol 161 (906) ◽  
pp. 353-366 ◽  
Keyword(s):  
Partial Differential Equations ◽  
Numerical Methods ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Formal Solution ◽  
Partial Differential ◽  
Numerical Work ◽  
Independent Variables ◽  
Wide Range ◽  
Technical Interest

The development of mechanical means of evaluating solutions of ordinary differential equations, in the form of the differential analyser of Dr. Bush (Bush 1931; Hartree 1935), has made it feasible to undertake the investigation of many problems of scientific and technical interest leading to differential equations which have no convenient formal solution, and which are too elaborate, or for which the range of solutions required is too extensive, for calculation of the solutions by numerical methods to be practicable. The practical success of this machine, and the wide range of equations to which it can be applied, have led to the hope that it may be found possible to apply it to partial differential equations, which are usually regarded as less amenable to numerical methods than ordinary equations. The present paper gives one way of applying it to such equations in two independent variables with certain types of boundary conditions. As will appear, the possibility of applying this method depends more on the form of the boundary conditions than on the exact form of the equations. The method is particularly suited to the differential analyser, though it is also practicable for numerical work.

Titchmarsh's theorem of Hankel transform

2019 ◽  
pp. 11-24
Author(s):  
Mohamed-Ahmed Boudref
Keyword(s):  
Number Theory ◽  
Data Analysis ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Lipschitz Condition ◽  
Analytic Number Theory ◽  
Hankel Transform ◽  
Partial Differential ◽  
Analytic Number ◽  
Bessel Transform

Hankel transform (or Fourier-Bessel transform) is a fundamental tool in many areas of mathematics and engineering, including analysis, partial differential equations, probability, analytic number theory, data analysis, etc. In this article, we prove an analog of Titchmarsh's theorem for the Hankel transform of functions satisfying the Hankel-Lipschitz condition.

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