scholarly journals Direct integration of systems of linear differential and difference equations

Filomat ◽  
2019 ◽  
Vol 33 (5) ◽  
pp. 1453-1461
Author(s):  
Syrgak Kydyraliev ◽  
Anarkul Urdaletova
Keyword(s):  
Differential Equations ◽  
Difference Equations ◽  
Main Idea ◽  
Euler Method ◽  
Linear Differential Equations ◽  
Combination Method ◽  
Direct Integration ◽  
Matrix Eigenvalues ◽  
Linear Differential ◽  
System Variables

Traditionally the Euler method is used for solving systems of linear differential equations. The method is based on the use of eigenvalues of a system?s coefficients matrix. Another method to solve those systems is the D?Alembert integrable combination method. In this paper, we present a new method for solving systems of linear differential and difference equations. The main idea of the method is using the coefficients matrix eigenvalues to find integrable combinations of system variables. This method is particularly advantageous when nonhomogeneous systems are considered.

scholarly journals Efficient quantum algorithm for dissipative nonlinear differential equations

2021 ◽  
Vol 118 (35) ◽  
pp. e2026805118
Author(s):  
Jin-Peng Liu ◽  
Herman Øie Kolden ◽  
Hari K. Krovi ◽  
Nuno F. Loureiro ◽  
Konstantina Trivisa ◽  
...  
Keyword(s):  
Differential Equations ◽  
Quantum Algorithms ◽  
Nonlinear Differential Equations ◽  
Quantum Algorithm ◽  
Euler Method ◽  
Linear Differential Equations ◽  
Dimensional System ◽  
Worst Case ◽  
Potential Applications ◽  
Linear Differential

Nonlinear differential equations model diverse phenomena but are notoriously difficult to solve. While there has been extensive previous work on efficient quantum algorithms for linear differential equations, the linearity of quantum mechanics has limited analogous progress for the nonlinear case. Despite this obstacle, we develop a quantum algorithm for dissipative quadratic n-dimensional ordinary differential equations. Assuming R<1, where R is a parameter characterizing the ratio of the nonlinearity and forcing to the linear dissipation, this algorithm has complexity T2q poly(log⁡T,log⁡n,log⁡1/ϵ)/ϵ, where T is the evolution time, ϵ is the allowed error, and q measures decay of the solution. This is an exponential improvement over the best previous quantum algorithms, whose complexity is exponential in T. While exponential decay precludes efficiency, driven equations can avoid this issue despite the presence of dissipation. Our algorithm uses the method of Carleman linearization, for which we give a convergence theorem. This method maps a system of nonlinear differential equations to an infinite-dimensional system of linear differential equations, which we discretize, truncate, and solve using the forward Euler method and the quantum linear system algorithm. We also provide a lower bound on the worst-case complexity of quantum algorithms for general quadratic differential equations, showing that the problem is intractable for R≥2. Finally, we discuss potential applications, showing that the R<1 condition can be satisfied in realistic epidemiological models and giving numerical evidence that the method may describe a model of fluid dynamics even for larger values of R.

scholarly journals Further study on solutions of some non-linear homogeneous differential equations in connection to Brück conjecture

2021 ◽  
Vol 29 (2) ◽  
Author(s):  
DILIP CHANDRA PRAMANIK ◽  
KAPIL ROY
Keyword(s):  
Differential Equations ◽  
Difference Equations ◽  
Linear Differential Equations ◽  
Linear Complex ◽  
Non Linear ◽  
Complex Differential Equations ◽  
Sharing Function ◽  
Linear Differential ◽  
Brück Conjecture

In this paper, using the theory of complex differential equations, we study the solution of some non-linear complex differential equations in connection to Brück conjecture which generalized some earlier results due to Pramanik, D. C. and Biswas, M., On solutions of some non-linear differential equations in connection to Bruck conjecture, Tamkang J. Math., 48 (2017), No. 4, 365–375; and Wang, H., Yang, L-Z. and Xu, H-Y., On some complex differential and difference equations concerning sharing function, Adv. Diff. Equ., 2014, 2014:274.

scholarly journals A New Scheme Using Cubic B-Spline to Solve Non-Linear Differential Equations Arising in Visco-Elastic Flows and Hydrodynamic Stability Problems

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1078 ◽  
Author(s):  
Asifa Tassaddiq ◽  
Aasma Khalid ◽  
Muhammad Nawaz Naeem ◽  
Abdul Ghaffar ◽  
Faheem Khan ◽  
...  
Keyword(s):  
Differential Equations ◽  
Linear Equations ◽  
Main Idea ◽  
Linear Differential Equations ◽  
System Of Linear Equations ◽  
B Spline ◽  
Stability Problems ◽  
Non Linear ◽  
Linear Differential ◽  
Magnetic Stability

This study deals with the numerical solution of the non-linear differential equations (DEs) arising in the study of hydrodynamics and hydro-magnetic stability problems using a new cubic B-spline scheme (CBS). The main idea is that we have modified the boundary value problems (BVPs) to produce a new system of linear equations. The algorithm developed here is not only for the approximation solutions of the 10th order BVPs but also estimate from 1st derivative to 10th derivative of the exact solution as well. Some examples are illustrated to show the feasibility and competence of the proposed scheme.

On the Operational Solution of Linear Finite Difference Equations

1931 ◽  
Vol 27 (1) ◽  
pp. 26-36 ◽  
Author(s):  
L. M. Milne-Thomson
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Difference Equations ◽  
Linear Differential Equations ◽  
Operational Method ◽  
Finite Difference Equations ◽  
Operational Solution ◽  
Linear Differential

The application of the operational method of Oliver Heaviside to the solution of linear differential equations has been fully described in a recent Cambridge Tract by Dr H. Jeffreys.

The function and associated polynomials

1951 ◽  
Vol 47 (2) ◽  
pp. 309-314 ◽  
Author(s):  
D. F. Lawden
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
Difference Equations ◽  
Linear Differential Equations ◽  
Linear Difference Equations ◽  
Transform Method ◽  
Associated Polynomials ◽  
The Laplace Transform ◽  
Linear Differential ◽  
Image Position

A transform method for the solution of linear difference equations, analogous to the method of the Laplace transform in the field of linear differential equations, has been described by Stone (1). The transform u(z) of a sequence un is defined by the equation

Sign in / Sign up

Export Citation Format

Share Document