scholarly journals Solutions of nonstandard initial value problems for a first order ordinary differential equation

1989 ◽  
Vol 2 (4) ◽  
pp. 225-237 ◽  
Author(s):  
M. Venkatesulu ◽  
P. D. N. Srinivas
Keyword(s):  
Functional Equation ◽  
Initial Conditions ◽  
Local Existence ◽  
Existence Theory ◽  
Initial Value ◽  
Order Ordinary Differential Equation ◽  
First Order ◽  
Long Time ◽  
General Existence ◽  
Physical Phenomena

Differential equations of the form y′=f(t,y,y′), where f is not necessarily linear in its arguments, represent certain physical phenomena and have been known to mathematicians for quite a long time. But a fairly general existence theory for solutions of the above type of problems does not exist because the (nonstandard) initial value problem y′=f(t,y,y′), y(t0)=y0 does not permit an equivalent integral equation of the conventional form. Hence, our aim here is to present a systematic study of solutions of the NSTD IVPs mentioned above.First, we establish the equivalence of the NSTD IVP with a functional equation and prove the local existence of a unique solution of the NSTD IVP via the functional equation. Secondly, we prove the continuous dependence of the solutions on initial conditions and parameters. Finally, we prove a global existence result and present an example to illustrate the theory.

scholarly journals Thermomechanically coupled modelling for land-terminating glaciers: a comparison of two-dimensional, first-order and three-dimensional, full-Stokes approaches

2015 ◽  
Vol 61 (228) ◽  
pp. 702-712 ◽  
Author(s):  
Tong Zhang ◽  
Lili Ju ◽  
Wei Leng ◽  
Stephen Price ◽  
Max Gunzburger
Keyword(s):  
Initial Conditions ◽  
Three Dimensional ◽  
Integration Time ◽  
Two Dimensional ◽  
Shape Factors ◽  
Coupled Modelling ◽  
First Order ◽  
Glacier Changes ◽  
Basal Sliding ◽  
Long Time

AbstractFor many regions, glacier inaccessibility results in sparse geometric datasets for use as model initial conditions (e.g. along the central flowline only). In these cases, two-dimensional (2-D) flowline models are often used to study glacier dynamics. Here we systematically investigate the applicability of a 2-D, first-order Stokes approximation flowline model (FLM), modified by shape factors, for the simulation of land-terminating glaciers by comparing it with a 3-D, ‘full’-Stokes ice-flow model (FSM). Based on steady-state and transient, thermomechanically uncoupled and coupled computational experiments, we explore the sensitivities of the FLM and FSM to ice geometry, temperature and forward model integration time. We find that, compared to the FSM, the FLM generally produces slower horizontal velocities, due to simplifications inherent to the FLM and to the underestimation of the shape factor. For polythermal glaciers, those with temperate ice zones, or when basal sliding is important, we find significant differences between simulation results when using the FLM versus the FSM. Over time, initially small differences between the FLM and FSM become much larger, particularly near cold/temperate ice transition surfaces. Long time integrations further increase small initial differences between the two models. We conclude that the FLM should be applied with caution when modelling glacier changes under a warming climate or over long periods of time.

scholarly journals Generalized (ψ,α,β)—Weak Contractions for Initial Value Problems

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 266 ◽  
Author(s):  
Piyachat Borisut ◽  
Poom Kumam ◽  
Vishal Gupta ◽  
Naveen Mani
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Initial Value Problems ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Weak Contraction ◽  
Initial Value ◽  
Order Ordinary Differential Equation ◽  
Ordered Metric Spaces ◽  
First Order

A class of generalized ( ψ , α , β ) —weak contraction is introduced and some fixed-point theorems in a framework of partially ordered metric spaces are proved. The main result of this paper is applied to a first-order ordinary differential equation to find its solution.

Numerical Solution of the Flow Between Two Disks

1997 ◽  
Author(s):  
Wahid S. Ghaly ◽  
Georgios H. Vatistas
Keyword(s):  
Shooting Method ◽  
Static Pressure ◽  
Numerical Solutions ◽  
Kutta Method ◽  
Initial Value ◽  
Third Order ◽  
Order Ordinary Differential Equation ◽  
First Order ◽  
Pressure Distributions ◽  
Runge Kutta Method

Abstract This paper deals with the numerical solutions of converging and diverging flows, between two disks. The results are obtained by solving a nonlinear third order ordinary differential equation using a modified shooting method. The governing equation is written as a system of three nonlinear first order ODE’s and the resulting system is solved as an initial value problem via the Runge-Kutta method. The results are given in terms of velocity profiles and static pressure distributions. These are compared with previously reported experimental data obtained by others.

scholarly journals On a multipoint nonlocal initial value problem for a singularly-perturbed first-order ODE

2019 ◽  
Vol 2019 (1) ◽  
pp. 84-103
Author(s):  
Dovlet M. Dovletov
Keyword(s):  
Initial Value Problem ◽  
A Priori ◽  
A Priori Estimate ◽  
Singularly Perturbed ◽  
Positive Parameter ◽  
Priori Estimate ◽  
Initial Value ◽  
Order Ordinary Differential Equation ◽  
Differential Problem ◽  
First Order

Abstract A linear first order ordinary differential equation (ODE) with a positive parameter ɛ and a multipoint nonlocal initial value condition (NLIVC) is considered. The existence of a classical solution of the multipoint nonlocal initial value problem (NLIVP) is proved. A uniform on ɛ a priori estimate and asymptotic expansion of smooth solution is obtained. The differential problem with integral kind of NLIVC is considered and reduced to appropriate multipoint NLIVP.

scholarly journals Existence criteria for singular initial value problems with sign changing nonlinearities

2001 ◽  
Vol 7 (6) ◽  
pp. 503-524 ◽  
Author(s):  
Ravi P. Agarwal ◽  
Donal O'Regan ◽  
V. Lakshmikantham
Keyword(s):  
Initial Value Problems ◽  
Existence Theory ◽  
Initial Value ◽  
General Existence ◽  
Existence Criteria

A general existence theory is presented for initial value problems where our nonlinearity may be singular in its dependent variable and may also change sign.

scholarly journals Numerical solutions of nonstandard first order initial value problems

1992 ◽  
Vol 5 (1) ◽  
pp. 69-82 ◽  
Author(s):  
M. Venkatesulu ◽  
P. D. N. Srinivasu
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Unique Solution ◽  
Initial Value Problem ◽  
Initial Value Problems ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Initial Value ◽  
First Order ◽  
Physical Phenomena

Differential equations of the form y′=f(t,y,y′), where f is not necessarily linear in its arguments, represent certain physical phenomena and are known for quite some time. The well known Clairut's and Chrystal's equations fall into this category. Earlier, we established the existence of a (unique) solution of the nonstandard initial value problem (NSTD IV P) y′=f(t,y,y′), y(t0)=y0 under certain natural hypotheses on f. In this paper we present some first order convergent numerical methods for finding the approximate solutions of the NST D I V Ps.

scholarly journals Modified Wiener equations

2001 ◽  
Vol 27 (6) ◽  
pp. 347-356 ◽  
Author(s):  
Will Watkins
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Initial Value Problem ◽  
Initial Conditions ◽  
Functional Differential Equations ◽  
Higher Order ◽  
Initial Value ◽  
Order Ordinary Differential Equation ◽  
First Case ◽  
Functional Differential

This paper is concerned with a class of functional differential equations whose argument transforms are involutions. In contrast to the earlier works in this area, which have used only involutions with a fixed point, we also admit involutions without a fixed point. In the first case, an initial value problem for a differential equation with involution is reduced to an initial value problem for a higher order ordinary differential equation. In our case, either two initial conditions or two boundary conditions are necessary for a solution; the equation is then reduced to a boundary value problem for a higher order ODE.

scholarly journals Stability of solutions of a nonstandard ordinary differential system by Lyapunov's second method

1991 ◽  
Vol 4 (3) ◽  
pp. 211-224
Author(s):  
M. Venkatesulu ◽  
P. D. N. Srinivasu
Keyword(s):  
Unique Solution ◽  
Differential System ◽  
Initial Value Problem ◽  
Stability Of Solutions ◽  
Initial Value ◽  
First Order ◽  
Ordinary Differential System ◽  
Lyapunov’S Second Method ◽  
The Stability ◽  
Physical Phenomena

Differential equations of the form y′=f(t,y,y′) where f is not necessarily linear in its arguments represent certain physical phenomena and are known for quite some time. The well known Clairut's and Chrystal's equations fall into this category. Earlier we established the existence of a (unique) solution of the nonstandard initial value problem y′=f(t,y,y′), y(t0)=y0 under certain natural hypotheses on f. In this paper, we studied the stability of solutions of a nonstandard first order ordinary differential system.

scholarly journals Perturbation Solutions to Fifth Order Over-damped Nonlinear Systems

2019 ◽  
pp. 1-11
Author(s):  
Harun-Or- Roshid ◽  
M. Zulfikar Ali ◽  
Pinakee Dey ◽  
M. Ali Akbar
Keyword(s):  
Nonlinear Systems ◽  
Initial Conditions ◽  
Approximate Solutions ◽  
First Order ◽  
Engineering Problems ◽  
Kbm Method ◽  
Fifth Order ◽  
Physical Phenomena ◽  
Good Agreement ◽  
Perturbation Solutions

Fifth order over-damp nonlinear differential systems can be used to describe many engineering problems and physical phenomena occur in the nature. In this article, the Krylov-Bogoliubov-Mitropolskii (KBM) method has been extended to investigate the solution of a certain fifth order over-damp nonlinear systems and desired result has been found. The implementation of the presented method is illustrated by an example. The first order analytical approximate solutions obtained by the method for different initial conditions show a good agreement with those obtains by numerical method.

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