Spline Interpolation Method Chemical Engineering Majors Authors Autar

Spline Interpolation Method Chemical Engineering Majors Authors: Autar Kaw, Jai Paul http: //numericalmethods. eng. usf. edu Transforming Numerical Methods Education for STEM Undergraduates http: //numericalmethods. eng. usf. edu 1

Spline Method of Interpolation http: //numericalmethods. eng. usf. edu

What is Interpolation ? Given (x 0, y 0), (x 1, y 1), …… (xn, yn), find the value of ‘y’ at a value of ‘x’ that is not given. 3 lmethods. eng. usf. edu http: //numerica

Interpolants Polynomials are the most common choice of interpolants because they are easy to: Evaluate Differentiate, and Integrate. 4 lmethods. eng. usf. edu http: //numerica

Why Splines ? 5 lmethods. eng. usf. edu http: //numerica

Why Splines ? 6 Figure : Higher order polynomial interpolation is a bad ideahttp: //numerica lmethods. eng. usf. edu

Linear Interpolation 7 lmethods. eng. usf. edu http: //numerica

Linear Interpolation (contd) 8 lmethods. eng. usf. edu http: //numerica

Example To find how much heat is required to bring a kettle of water to its boiling point, you are asked to calculate the specific heat of water at 61 °C. The specific heat of water is given as a function of time in Table 1. Use linear spline interpolation to determine the value of the specific heat at T = 61°C. Table 1 Specific heat of water as a function of temperature. 9 Temperature, Specific heat, 22 42 52 82 100 4181 4179 4186 4199 4217 Figure 2 Specific heat of water vs. temperature. http: //numerica lmethods. eng. usf. edu

Linear Interpolation 10 lmethods. eng. usf. edu http: //numerica

Quadratic Interpolation 11 lmethods. eng. usf. edu http: //numerica

Quadratic Interpolation (contd) 12 lmethods. eng. usf. edu http: //numerica

Quadratic Splines (contd) 13 lmethods. eng. usf. edu http: //numerica

Quadratic Splines (contd) 14 lmethods. eng. usf. edu http: //numerica

Quadratic Splines (contd) 15 lmethods. eng. usf. edu http: //numerica

Example To find how much heat is required to bring a kettle of water to its boiling point, you are asked to calculate the specific heat of water at 61 °C. The specific heat of water is given as a function of time in Table 1. Use quadratic spline interpolation to determine the value of the specific heat at T = 61°C. Table 1 Specific heat of water as a function of temperature. 16 Temperature, Specific heat, 22 42 52 82 100 4181 4179 4186 4199 4217 Figure 2 Specific heat of water vs. temperature. http: //numerica lmethods. eng. usf. edu

Solution 17 lmethods. eng. usf. edu http: //numerica

Solution (contd) 18 lmethods. eng. usf. edu http: //numerica

Solution (contd) 19 lmethods. eng. usf. edu http: //numerica

Solution (contd) 20 lmethods. eng. usf. edu http: //numerica

Solution (contd) 21 lmethods. eng. usf. edu http: //numerica

Solution (contd) 22 lmethods. eng. usf. edu http: //numerica

Better Estimate 23 lmethods. eng. usf. edu http: //numerica

Better Estimate 24 lmethods. eng. usf. edu http: //numerica

Better Estimate 25 lmethods. eng. usf. edu http: //numerica

Additional Resources For all resources on this topic such as digital audiovisual lectures, primers, textbook chapters, multiple-choice tests, worksheets in MATLAB, MATHEMATICA, Math. Cad and MAPLE, blogs, related physical problems, please visit http: //numericalmethods. eng. usf. edu/topics/spline_met hod. html

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