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Dr. Jie Zou PHY 33201 Chapter 7 Numerical Differentiation: 1 Lecture (I) 1 Ref: “Applied Numerical Methods with MATLAB for Engineers and Scientists”, Steven Chapra, 2 nd ed., McGraw Hill, 2008.

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Dr. Jie Zou PHY 33202 Outline Engineering and scientific applications Definition of the derivative-a review of calculus Numerical differentiation (1) Basic finite-difference approximations for both the 1 st and 2 nd derivatives (I) Forward difference (II) Backward difference (III) Centered difference (2) Higher-accuracy finite-difference approximations (Lecture II)

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Dr. Jie Zou PHY 33203 Engineering and scientific applications The mathematical models of most engineering and science problems involve differentiation. Examples: Mechanics: Newton’s 2 nd law (1D), F = md 2 x/dt 2. Heat conduction: Fourier’s law (1D), q = - kdT/dx. Electrical conduction: Ohm’s law (1D), J = - dV/dx. When do we need apply methods of numerical differentiation? When it is difficult to differentiate analytically; When the function is unknown and defined only by measurement at discrete points. Heat flow and temperature gradient (Ref. Fig. 19.2)

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4 Definition of the derivative (First) derivative: The rate of change of a dependent variable, such as y, with respect to an independent variable, such as x. Graphical representation Difference approximation Definition of the derivative

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Dr. Jie Zou PHY 33205 Definition of the derivative (cont.) Second derivative: The derivative of the first derivative. Mathematical definition: The 1 st derivative tells us the slope of the tangent to the curve at a given point. The 2 nd derivative tells us how fast the slope is changing. It is also called the curvature. Partial derivatives of f(x, y):

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Dr. Jie Zou PHY 33206 Basic finite-difference approximations Taylor series expansion of f(x) about a point x i : Evaluating f(x) at point x i+1, we have: Truncating the series after the first derivative term, we have: Forward difference approx. of the 1 st derivative: h: The step size - the length of the interval over which the approximation is made; h = x i+1 – x i.

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7 Basic finite-difference approximations (cont.) Backward difference approx. of the 1 st derivative: Centered difference approx. of the 1 st derivative: Note: The truncation error in the Centered difference approx. is of the order of h 2 ; the truncation error in the Forward and Backward difference approx. is of the order of h. Centered difference approx. is more accurate than the Forward or Backward difference approx.

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Dr. Jie Zou PHY 33208 Basic finite-difference approximations (cont.) Finite-difference approximations of the 2 nd derivatives: Second forward finite difference: Second backward finite difference: Second centered finite difference:

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Dr. Jie Zou PHY 33209 Example (Ref. example 4.4) Use a forward, backward, and centered difference approximation to estimate the 1 st derivative of f(x) = -0.1x 4 – 0.15x 3 – 0.5x 2 – 0.25x + 1.2 at x = 0.5 using a step size h = 0.5. Repeat the computation using h = 0.25. Find the percent error of the estimation relative to the true value.

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Dr. Jie Zou PHY 332010 Results (Example 4.4) h=0.5Backward O(h) Centered O(h 2 ) Forward O(h) Estimate-0.55-1.45 |t||t| 39.7%9.6%58.9% h=0.25Backward O(h) Centered O(h 2 ) Forward O(h) Estimate-0.714-0.934-1.155 |t||t| 21.7%2.4%26.5% True value: f’(0.5) = -0.9125

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