Numerical solution of two differential-difference equations of analytic theory of numbers

1969 ◽  
pp. 179-187 ◽  
Author(s):  
Mohan Lal ◽  
Paul Gillard
Keyword(s):  
Numerical Solution ◽  
Difference Equations ◽  
Analytic Theory ◽  
Theory Of Numbers

Some Applications of Fourier series in the Analytic Theory of Numbers

1928 ◽  
Vol 24 (4) ◽  
pp. 585-596 ◽  
Author(s):  
L. J. Mordell
Keyword(s):  
Fourier Series ◽  
Class Number ◽  
Lattice Point ◽  
Zeta Functions ◽  
Gauss Sums ◽  
Analytic Theory ◽  
Quadratic Fields ◽  
Lattice Point Problems ◽  
Theory Of Numbers

It is a familiar fact that an important part is played in the Analytic Theory of Numbers by Fourier series. There are, for example, applications to Gauss' sums, to the zeta functions, to lattice point problems, and to formulae for the class number of quadratic fields.

Ω theorems for the complex divisor function

1994 ◽  
Vol 115 (1) ◽  
pp. 145-157
Author(s):  
R. R. Hall
Keyword(s):  
Analytic Theory ◽  
Divisor Function ◽  
Image Position ◽  
Theory Of Numbers

This paper is a sequel to [6] and concerns the complex divisor functionwhich has had a number of applications in the analytic theory of numbers. Thus Hooley's Δ-function [8] defined bysatisfies the inequalityand Erdös' τ+-function, defined bysatisfies

scholarly journals A New Algorithm for Solving Terminal Value Problems of q-Difference Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-7
Author(s):  
Yong-Hong Fan ◽  
Lin-Lin Wang
Keyword(s):  
Exact Solution ◽  
Numerical Methods ◽  
Error Estimate ◽  
Numerical Solution ◽  
Numerical Method ◽  
Difference Equations ◽  
Initial Value Problems ◽  
Convergence Speed ◽  
Initial Value ◽  
Delta Derivatives

We propose a new algorithm for solving the terminal value problems on a q-difference equations. Through some transformations, the terminal value problems which contain the first- and second-order delta-derivatives have been changed into the corresponding initial value problems; then with the help of the methods developed by Liu and H. Jafari, the numerical solution has been obtained and the error estimate has also been considered for the terminal value problems. Some examples are given to illustrate the accuracy of the numerical methods we proposed. By comparing the exact solution with the numerical solution, we find that the convergence speed of this numerical method is very fast.

Solution of the Immiscible Fluid Flow Simulation Equations

1969 ◽  
Vol 9 (02) ◽  
pp. 155-169 ◽  
Author(s):  
E.A. Breitenbach ◽  
D.H. Thurnau ◽  
H.K. Van Poolen
Keyword(s):  
Finite Difference ◽  
Numerical Solution ◽  
Difference Equations ◽  
Computing Time ◽  
Flow Simulation ◽  
Computational Technique ◽  
Immiscible Fluid ◽  
Finite Difference Equations ◽  
Gauss Elimination ◽  
Point Form

American Institute of Mining, Metallurgical and Petroleum Engineers, Inc. Abstract This paper presents the methods used to solve the finite difference equations which we developed in a companion paper (1). Various possible methods of solution are discussed. Experience has narrowed the numb of suitable numerical methods that are practical to three: Gauss elimination, successive practical to three: Gauss elimination, successive overrelaxation, and the iterative alternating direction implicit process. The final sections of the paper are devoted to a presentation of computational technique which are vital to actual use of each of the above-mentioned methods. FINITE DIFFERENCE EQUATIONS, THE MATRIX AND DEFINITIONS The final finite difference equation for pressure developed in Reference (1) is: pressure developed in Reference (1) is: ..........................................(1) All the terms are defined in the paper. Here, however, we have dropped the subscript denoting the pressure, p, as an oil pressure. Further breakdown requires definition of the numerical solution to be used. This paper describes the breakdown and solution processes most often used in the MUFFS program. Sufficient detail is given so that computer programming can be done. Contrary to popular opinion, economic simulation has been found to require the development of several solution methods, rather than relying on a single one. This requires that the computer subprogram for generating coefficients (A's and O's) be written as a distinct, separate entity to supply the coefficients in Equation (1). Furthermore, it is necessary to be able to obtain these coefficients automatically in column-by-column, row-by-raw, or point-by-point form, in any order required by a point-by-point form, in any order required by a numerical solution. Columns, rows, and points refer to the columns, rows, and points of the finite difference grid. A program that can generate coefficients in several forms is a simple but important concept, for it allows the easy insertion and modification of experimental methods. The computing inefficiencies that may be incurred within a general coefficient generator are small in comparison to the computing time saved by using the fastest of several solution techniques.

Partition theory and thermodynamics of multi-dimensional oscillator assemblies

1951 ◽  
Vol 47 (3) ◽  
pp. 591-601 ◽  
Author(s):  
V. S. Nanda
Keyword(s):  
Generating Function ◽  
Statistical Thermodynamics ◽  
Analytic Theory ◽  
Close Similarity ◽  
Thermodynamic Approach ◽  
Large Integer ◽  
Partition Problem ◽  
Partition Theory ◽  
Theory Of Numbers

The close similarity between the basic problems in statistical thermodynamics and the partition theory of numbers is now well recognized. In either case one is concerned with partitioning a large integer, under certain restrictions, which in effect means that the ‘Zustandsumme’ of a thermodynamic assembly is identical with the generating function of partitions appropriate to that assembly. The thermodynamic approach to the partition problem is of considerable interest as it has led to generalizations which so far have not yielded to the methods of the analytic theory of numbers. An interesting example is provided in a recent paper of Agarwala and Auluck (1) where the Hardy Ramanujan formula for partitions into integral powers of integers is shown to be valid for non-integral powers as well.

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