Construction of Five-step Continuous Block General Method for the Solution of Ordinary Differential Equations

2015 ◽  
Vol 4 (6) ◽  
pp. 574-584
Author(s):  
D. Raymond ◽  
J. Donald ◽  
J. Oladunjoye ◽  
A. Lydia
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Continuous Block ◽  
General Method

scholarly journals Using Ordinary Differential Equations to Explore Cancer-Immune Dynamics and Tumor Dormancy

2016 ◽  
Author(s):  
Kathleen P. Wilkie ◽  
Philip Hahnfeldt ◽  
Lynn Hlatky
Keyword(s):  
Mathematical Modeling ◽  
Immune Response ◽  
Differential Equations ◽  
Tumor Growth ◽  
Ordinary Differential Equations ◽  
Systemic Disease ◽  
Growth Dynamics ◽  
Tumor Dormancy ◽  
Cytotoxic Action ◽  
General Method

AbstractCancer is not solely a disease of the genome, but is a systemic disease that affects the host on many functional levels, including, and perhaps most notably, the function of the immune response, resulting in both tumor-promoting inflammation and tumor-inhibiting cytotoxic action. The dichotomous actions of the immune response induce significant variations in tumor growth dynamics that mathematical modeling can help to understand. Here we present a general method using ordinary differential equations (ODEs) to model and analyze cancer-immune interactions, and in particular, immune-induced tumor dormancy.

scholarly journals The Laplace transform as an alternative general method for solving linear ordinary differential equations

2021 ◽  
Vol 1 (4) ◽  
pp. 309
Author(s):  
William Guo
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Laplace Transform ◽  
Ordinary Differential Equations ◽  
Case Studies ◽  
Initial Conditions ◽  
Linear Ordinary Differential Equations ◽  
The Laplace Transform ◽  
Linear Odes ◽  
General Method

<p style='text-indent:20px;'>The Laplace transform is a popular approach in solving ordinary differential equations (ODEs), particularly solving initial value problems (IVPs) of ODEs. Such stereotype may confuse students when they face a task of solving ODEs without explicit initial condition(s). In this paper, four case studies of solving ODEs by the Laplace transform are used to demonstrate that, firstly, how much influence of the stereotype of the Laplace transform was on student's perception of utilizing this method to solve ODEs under different initial conditions; secondly, how the generalization of the Laplace transform for solving linear ODEs with generic initial conditions can not only break down the stereotype but also broaden the applicability of the Laplace transform for solving constant-coefficient linear ODEs. These case studies also show that the Laplace transform is even more robust for obtaining the specific solutions directly from the general solution once the initial values are assigned later. This implies that the generic initial conditions in the general solution obtained by the Laplace transform could be used as a point of control for some dynamic systems.</p>

scholarly journals Four-Step One Hybrid Block Methods for Solution of Fourth Derivative Ordinary Differential Equations

2021 ◽  
pp. 1-10
Author(s):  
Raymond, Dominic ◽  
Skwame, Yusuf ◽  
Adiku, Lydia
Keyword(s):  
Approximate Solution ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Multistep Method ◽  
Linear Multistep Method ◽  
Block Method ◽  
Step Method ◽  
Block Methods ◽  
Fourth Derivative ◽  
Continuous Block

We consider developing a four-step one offgrid block hybrid method for the solution of fourth derivative Ordinary Differential Equations. Method of interpolation and collocation of power series approximate solution was used as the basis function to generate the continuous hybrid linear multistep method, which was then evaluated at non-interpolating points to give a continuous block method. The discrete block method was recovered when the continuous block was evaluated at all step points. The basic properties of the methods were investigated and said to be converge. The developed four-step method is applied to solve fourth derivative problems of ordinary differential equations from the numerical results obtained; it is observed that the developed method gives better approximation than the existing method compared with.

scholarly journals Tides in oceans bounded by meridians

1936 ◽  
Vol 235 (753) ◽  
pp. 273-342 ◽  
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Infinite Sequence ◽  
Double Sequence ◽  
Algebraic Equations ◽  
Dynamical Equations ◽  
Mathematical Problems ◽  
Single Sequence ◽  
General Method

This is the first part of a series of papers by A. T. Doodson and myself, in which we intend to publish certain investigations which we have been carrying out intermittently for some years. In 1916 I published an account* of a general method of treating the dynamical equations of the tides in which the ordinary differential equations were transformed into an infinite sequence of algebraic equations. One of the chief features of the treatment is that an attempt was made to deal rigorously with questions of convergence. At that time the determination of the tides in a flat rectangular sea, a flat sectorial sea, and an ocean bounded by two meridians constituted mathematical problems which were completely unsolved, and I pointed out that for basins of these shapes and of uniform depth the coefficients in my algebraic equations could easily be evaluated. It is a disadvantage of the method, however, as applied to these systems, that the algebraic equations are naturally arranged in a double sequence and not in a single sequence.

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