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Chapter 1 Some Codes of the Software Mathematica When we study the nonlinear waves of some partial differential equations£¬ a mathematical software called ¡°Mathematica¡± is often used to perform qualitative analysis and quantitative simulation. In this chapter£¬ we list some useful functions and their corresponding codes which will be employed in the following chapters. 1. Use ¡°Solve¡± to solve equations Example 1.1 Solve the following equation (1.1) Solution The codes and output result are as follows: Example 1.2 Solve a set of equations (1.2) Solution The codes and output result are as follows. 2. Use ¡°NSolve¡± to solve equations Example 1.3 Solve the following equation (1.3) Solution The codes and output result are as follows: NSolve£¬ Example 1.4 Solve a set of equations (1.4) Solution The codes and output result are as follows: NSolve. 3. Use ¡°Factor¡± to perform factorization of polynomials Example 1.5 Factorize the following polynomial (1.5) Solution The codes and output result are as follows: Factor Example 1.6 Factorize the following polynomial (1.6) Solution The codes and output result are as follows: 4. Use ¡°Expand¡± to expand some polynomials Example 1.7 Expand the following polynomial (1.7) Solution The codes and output result are as follows: Expand Example 1.8 Expand the following polynomial (1.8) Solution The codes and output result are as follows: Expand. 5. Use ¡°Collect¡± to expand some polynomials Example 1.9 Expand the polynomial in terms of the power of x (1.9) Solution The codes and output result are as follows: Collect. Example 1.10 Expand the polynomial in terms of the power of y (1.10) Solution The codes and output result are as follows: Collect. 6. Use ¡°Series¡± to expand a function Example 1.11 At x = 0£¬ expand sin x up to the 10th-order term. Solution The codes and output result are as follows: Series[Sin[x]£¬ {x£¬ 0£¬ 10}] Example 1.12 At x = 1£¬ expand ln x up to the 6th-order term. Solution The codes and output result are as follows: Series[Log[x]£¬ {x£¬ 1£¬ 6}]. 7. Use ¡°Limit¡± to determine the limit of a sequence or function Example 1.13 Determine the limit of when n tends to positive infinity. Solution The codes and output result are as follows: Example 1.14 When x tends to zero£¬ determine the limit of the following function (1.11) Solution The codes and output result are as follows: 8. Use ¡°D¡± and ¡°Simplify¡± to calculate the derivative£¬ partial derivative and verify the correctness of a solution Example 1.15 Calculate the 1st-order and 3rd-order derivatives of the function f(x) respectively£¬ where (1.12) Solution The codes and output result are as follows: Example 1.16 Calculate the 1st-order partial derivative gx£¬ gy and 5th-order partial derivative gxxyyy of function g(x£¬ y) respectively£¬ where (1.13) Solution The codes and output result are as follows: Example 1.17 Test if function set x = .cos t and y = sin t is a set of solution of the following system (1.14) Solution The codes and output result are as follows: Example 1.18 Test if is a solution of the following equation (1.15) Solution The codes and output result are as follows: Simplify 9. Use ¡°Plot¡± to draw graph of the function y = f(x) Example 1.19 Draw the graphs of y = sin x and y = cos x respectively. Solution The codes and output result are as follows (see Figures 1.1 and 1.2): Plot Plot Example 1.20 Draw the graphs of y = cos x and y = sin x together. Solution The codes and output result are as follows (see Figure 1.3): Plot Figure 1.1 The graph of y = sin x Figure 1.2 The graph of y = cos x Figure 1.3 The graphs of y = cos x and y = sin x 10. Use ¡°ParametricPlot¡± to draw the curves with parametric expression Example 1.21 On the x¨Cy plane£¬ draw the curve with the expression (1.16) Solution The codes and output result are as follows (see Figure 1.4): Example 1.22 On the x¨Cy¨Cz space£¬ draw the curves with the expression (1.17) Solution The codes and output result are as follows (see Figure 1.5):