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 An Existence Theory for Gravity–Capillary Solitary Water Waves

Water Waves ◽  
2021 ◽  
Author(s):  
M. D. Groves
Keyword(s):  
Surface Tension ◽  
Water Waves ◽  
Applied Mathematics ◽  
Spatial Dynamics ◽  
Existence Theory ◽  
Water Wave Problem ◽  
Reduction Methods ◽  
Centre Manifold Reduction ◽  
Weak Surface ◽  
The One

AbstractIn the applied mathematics literature solitary gravity–capillary water waves are modelled by approximating the standard governing equations for water waves by a Korteweg-de Vries equation (for strong surface tension) or a nonlinear Schrödinger equation (for weak surface tension). These formal arguments have been justified by sophisticated techniques such as spatial dynamics and centre-manifold reduction methods on the one hand and variational methods on the other. This article presents a complete, self-contained account of an alternative, simpler approach in which one works directly with the Zakharov–Craig–Sulem formulation of the water-wave problem and uses only rudimentary fixed-point arguments and Fourier analysis.

scholarly journals Existence theory: Outline for a theory of social behaviour

2021 ◽  
pp. 1468795X2199824
Author(s):  
Patrick Baert ◽  
Marcus Morgan ◽  
Rin Ushiyama
Keyword(s):  
Formal Education ◽  
Sociological Theory ◽  
Forced Migration ◽  
Existence Theory ◽  
Power Dynamics ◽  
Structural Factors ◽  
Contemporary Society ◽  
Social Phenomena ◽  
New Approach ◽  
Cultural Expectations

This article introduces ‘existence theory’ as a new approach to sociological theory and research. Existence theory starts from the assumption that people organise their lives around a limited set of existential milestones. Cultural expectations are such that without the accomplishment of those milestones, individuals may experience their lives as incomplete. Examples of milestones can include the attainment of formal education, a lasting partnership and the creation of a family, but in general the milestones which are important to individuals and their precise articulation will depend on a variety of cultural and structural factors. The achievement of existential milestones often depends on that of other existential milestones, thereby producing what we call an ‘existential ladder’. The article also elaborates on the significance of ‘existential urgency’ in that, due to a variety of factors (some biological, some cultural and structural), there are time limits on when certain existential milestones ought to be achieved by. In contemporary society, we note that individuals seem to have more choice about which milestones are important to them and when they can be achieved, although we emphasise that this flexibility is unevenly distributed. This then provides a steppingstone towards an elaboration of the power dynamics and inequalities underlying both the experience and the achievement of existential milestones. Finally, this paper shows how existence theory helps to reflect on a variety of social phenomena of contemporary significance: populism in politics, forced migration, and the coronavirus pandemic.

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