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Introduction
Published in Vladimir A. Dobrushkin, Applied Differential Equations, 2022
Let us consider any region R of the xy-plane in which f(x,y) is a real, single-valued, continuous function. Then the differential equation y′=f(x,y) defines a direction field in the region R. A solution y=ϕ(x) of the given differential equation has the property that at every point its graph is tangent to the direction element at that point. The slope field provides useful qualitative information about the behavior of the solution even when you cannot solve it. Direction fields are common in physical applications, which we discuss in [16]. While slope fields prove their usefulness in qualitative analysis, they are open to several criticisms. The integral curves, being graphically obtained, are only approximations to the solutions without any knowledge of their accuracy and formulas.
Introduction
Published in Vladimir A. Dobrushkin, Applied Differential Equations, 2018
Let us consider any region R of the xy-plane in which f(x,y) is a real, single-valued, continuous function. Then the differential equation y'=f(x,y) defines a direction field in the region R. A solution y=ϕ(x) of the given differential equation has the property that at every point its graph is tangent to the direction element at that point. The slope field provides useful qualitative information about the behavior of the solution even when you cannot solve it. Direction fields are common in physical applications, which we discuss in [15]. While slope fields prove their usefulness in qualitative analysis, they are open to several criticisms. The integral curves, being graphically obtained, are only approximations to the solutions without any knowledge of their accuracy and formulas.
Introduction
Published in Vladimir A. Dobrushkin, Applied Differential Equations with Boundary Value Problems, 2017
Let us consider any region R of the xy-plane in which f(x,y) $ f(x, y) $ is a real, single-valued, continuous function. Then the differential equation y′=f(x,y) $ y^{'} = f(x, y) $ defines a direction field in the region R. A solution y = φ(x) of the given differential equation has the property that at every point its graph is tangent to the direction element at that point. The slope field provides useful qualitative information about the behavior of the solution even when you cannot solve it. Direction fields are common in physical applications, which we discuss in [14. While slope fields prove their usefulness in qualitative analysis, they are open to several criticisms. The integral curves, being graphically obtained, are only approximations to the solutions without any knowledge of their accuracy and formulas.
Qualitative aspects of differential equations in an inquiry-oriented course
Published in International Journal of Mathematical Education in Science and Technology, 2023
A central graphical approach emphasized in the IODE curriculum is the concept of slope fields. At any given point of the plane, the differential equation gives a value for the slope of the tangent to the solution curve passing through that point. A sketch of mini-tangents at various points of the plane constitutes the slope field (see a sample slope field in Figure 1). Starting at any given point (initial condition) and following the direction of the mini-tangents give a visual description of the solution through that point, a picture that portrays the increase and decrease of the solution, its concavity, its long term behaviour as . In the special case of an autonomous ODE , the phase line which is a one-dimensional projection of graphs of solution functions captures the entire qualitative behaviour of these solutions (see Figure 2).
Effect of sediment transport, flow depth and infiltration on soil moisture profiles in irrigation furrows
Published in Journal of Applied Water Engineering and Research, 2023
Kapil Rohilla, Sanjay Kumar, Satish Kumar
Irrigation furrows are widely used for surface irrigation, in which, water flows due to gravity. Sediment-laden water is often supplied directly from rivers or canals to the farm through irrigation furrows. Therefore, deposition takes place along the length of irrigation furrows. It may affect the movement of soil moisture in the subsurface. Several times, water from wells is applied for irrigation, which is sediment free. In such situations, the flow has high tractive force at the inlet of the irrigation channel. Consequently, erosion takes place upstream and deposition at downstream reaches of the irrigation channel. The sediment transport in an irrigation channel depends on the flow rate, infiltration rate, slope, field length, sediment size and sediment carrying capacity of the channel. As clear water flows in the irrigation channel, it picks up fine sediment, some of which get deposited around the wetted perimeter in downstream reaches of the irrigation channel. The side soil on the irrigation channel wets slowly due to the capillaries' action, and it is affected by tractive as well as gravitational force (Brown et al. 1988). Soil moisture is very important in many aspects, i.e. irrigation, agricultural, water resources and drought management. Soil water retention and hydraulic properties affect moisture movement in the vadose zone. These parameters are important for the simulation of moisture flow and solute transport dynamics and in the analysis of infiltration and drainage processes (Prasad et al. 2001). The moisture movement in subsurface of an irrigation channel depends on the flow depth, infiltration rate, slope, channel length and saturated hydraulic conductivity of the soil. Hence, it is imperative to have a thorough understanding of soil moisture profile and its quantification from irrigation furrows to provide advice on proper flow and sediment transport management.
A 3D-printable machine for conics and oblique trajectories
Published in International Journal of Mathematical Education in Science and Technology, 2022
Pietro Milici, Frédérique Plantevin, Massimo Salvi
Graphical representations of slope fields involve the simultaneous drawing of directions at many points in the plane. This representation is a static one; by machines, we can extend this idea to dynamic slope fields. Indeed, by considering a rod r with a point P marked on it, P freely moving on the plane, we can generate a slope field by a mechanism that links the inclination of r to the position of P. Such an inclination dynamically defines a slope field over the plane.