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The second derivative of a function f can be used to determine the concavity of the graph of f. [2] A function whose second derivative is positive is said to be concave up (also referred to as convex), meaning that the tangent line near the point where it touches the function will lie below the graph of the function.
See the example figure on the right. Appended to this nonlinear edge is an edge weight that is the second-order partial derivative of the nonlinear node in relation to its predecessors. This nonlinear edge is subsequently pushed down to further predecessors in such a way that when it reaches the independent nodes, its edge weight is the second ...
The Crank–Nicolson stencil for a 1D problem. The Crank–Nicolson method is based on the trapezoidal rule, giving second-order convergence in time.For linear equations, the trapezoidal rule is equivalent to the implicit midpoint method [citation needed] —the simplest example of a Gauss–Legendre implicit Runge–Kutta method—which also has the property of being a geometric integrator.
First-order means that only the first derivative of y appears in the equation, and higher derivatives are absent. Without loss of generality to higher-order systems, we restrict ourselves to first-order differential equations, because a higher-order ODE can be converted into a larger system of first-order equations by introducing extra variables.
where is a second-order elliptic operator (implying that must be positive; a case where = + is considered below). A system of partial differential equations for a vector can also be parabolic. For example, such a system is hidden in an equation of the form
for the nth derivative. When f is a function of several variables, it is common to use "∂", a stylized cursive lower-case d, rather than "D". As above, the subscripts denote the derivatives that are being taken. For example, the second partial derivatives of a function f(x, y) are: [6]
One application of higher-order derivatives is in physics. Suppose that a function represents the position of an object at the time. The first derivative of that function is the velocity of an object with respect to time, the second derivative of the function is the acceleration of an object with respect to time, [29] and the third derivative ...
Order Equation Application Reference Abel's differential equation of the first kind: 1 = + + + Class of differential equation which may be solved implicitly [1] Abel's differential equation of the second kind: 1