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Overview An Example Comparison to Actual Solution Conclusion Finite Difference Method Bernd Schr oder logo1 Bernd Schr oder Louisiana Tech University, College of Engineering and Science Finite Difference Method Overview An Example

  1. Overview An Example Comparison to Actual Solution Conclusion Discretize the Problem 1. Determine a grid x n , n = 0 ,..., N . 2. In the differential equation, replace y ′ ( x ) with y ( x + h ) − y ( x − h ) , 2 h replace y ′′ ( x ) with y ( x + h ) − 2 y ( x )+ y ( x − h ) . h 2 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  2. Overview An Example Comparison to Actual Solution Conclusion Discretize the Problem 1. Determine a grid x n , n = 0 ,..., N . 2. In the differential equation, replace y ′ ( x ) with y ( x + h ) − y ( x − h ) , 2 h replace y ′′ ( x ) with y ( x + h ) − 2 y ( x )+ y ( x − h ) . h 2 3. In the resulting equation, replace x with x n . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  3. Overview An Example Comparison to Actual Solution Conclusion Discretize the Problem 1. Determine a grid x n , n = 0 ,..., N . 2. In the differential equation, replace y ′ ( x ) with y ( x + h ) − y ( x − h ) , 2 h replace y ′′ ( x ) with y ( x + h ) − 2 y ( x )+ y ( x − h ) . h 2 3. In the resulting equation, replace x with x n . 4. In the resulting equation, replace y ( x n ) with y n logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  4. Overview An Example Comparison to Actual Solution Conclusion Discretize the Problem 1. Determine a grid x n , n = 0 ,..., N . 2. In the differential equation, replace y ′ ( x ) with y ( x + h ) − y ( x − h ) , 2 h replace y ′′ ( x ) with y ( x + h ) − 2 y ( x )+ y ( x − h ) . h 2 3. In the resulting equation, replace x with x n . 4. In the resulting equation, replace y ( x n ) with y n , replace y ( x n − h ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  5. Overview An Example Comparison to Actual Solution Conclusion Discretize the Problem 1. Determine a grid x n , n = 0 ,..., N . 2. In the differential equation, replace y ′ ( x ) with y ( x + h ) − y ( x − h ) , 2 h replace y ′′ ( x ) with y ( x + h ) − 2 y ( x )+ y ( x − h ) . h 2 3. In the resulting equation, replace x with x n . 4. In the resulting equation, replace y ( x n ) with y n , replace y ( x n − h ) = y ( x n − 1 ) with y n − 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  6. Overview An Example Comparison to Actual Solution Conclusion Discretize the Problem 1. Determine a grid x n , n = 0 ,..., N . 2. In the differential equation, replace y ′ ( x ) with y ( x + h ) − y ( x − h ) , 2 h replace y ′′ ( x ) with y ( x + h ) − 2 y ( x )+ y ( x − h ) . h 2 3. In the resulting equation, replace x with x n . 4. In the resulting equation, replace y ( x n ) with y n , replace y ( x n − h ) = y ( x n − 1 ) with y n − 1 , replace y ( x n + h ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  7. Overview An Example Comparison to Actual Solution Conclusion Discretize the Problem 1. Determine a grid x n , n = 0 ,..., N . 2. In the differential equation, replace y ′ ( x ) with y ( x + h ) − y ( x − h ) , 2 h replace y ′′ ( x ) with y ( x + h ) − 2 y ( x )+ y ( x − h ) . h 2 3. In the resulting equation, replace x with x n . 4. In the resulting equation, replace y ( x n ) with y n , replace y ( x n − h ) = y ( x n − 1 ) with y n − 1 , replace y ( x n + h ) = y ( x n + 1 ) with y n + 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  8. Overview An Example Comparison to Actual Solution Conclusion Discretize the Problem 1. Determine a grid x n , n = 0 ,..., N . 2. In the differential equation, replace y ′ ( x ) with y ( x + h ) − y ( x − h ) , 2 h replace y ′′ ( x ) with y ( x + h ) − 2 y ( x )+ y ( x − h ) . h 2 3. In the resulting equation, replace x with x n . 4. In the resulting equation, replace y ( x n ) with y n , replace y ( x n − h ) = y ( x n − 1 ) with y n − 1 , replace y ( x n + h ) = y ( x n + 1 ) with y n + 1 , replace x n with x 0 + nh logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  9. Overview An Example Comparison to Actual Solution Conclusion Discretize the Problem 1. Determine a grid x n , n = 0 ,..., N . 2. In the differential equation, replace y ′ ( x ) with y ( x + h ) − y ( x − h ) , 2 h replace y ′′ ( x ) with y ( x + h ) − 2 y ( x )+ y ( x − h ) . h 2 3. In the resulting equation, replace x with x n . 4. In the resulting equation, replace y ( x n ) with y n , replace y ( x n − h ) = y ( x n − 1 ) with y n − 1 , replace y ( x n + h ) = y ( x n + 1 ) with y n + 1 , replace x n with x 0 + nh , and replace h with its numerical value. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  10. Overview An Example Comparison to Actual Solution Conclusion Discretize the Problem 1. Determine a grid x n , n = 0 ,..., N . 2. In the differential equation, replace y ′ ( x ) with y ( x + h ) − y ( x − h ) , 2 h replace y ′′ ( x ) with y ( x + h ) − 2 y ( x )+ y ( x − h ) . h 2 3. In the resulting equation, replace x with x n . 4. In the resulting equation, replace y ( x n ) with y n , replace y ( x n − h ) = y ( x n − 1 ) with y n − 1 , replace y ( x n + h ) = y ( x n + 1 ) with y n + 1 , replace x n with x 0 + nh , and replace h with its numerical value. 5. Solve the resulting system of equations. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  11. Overview An Example Comparison to Actual Solution Conclusion Use Finite Differences with 8 Intervals to Solve the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  12. Overview An Example Comparison to Actual Solution Conclusion Use Finite Differences with 8 Intervals to Solve the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e h : = 1 − 0 = 1 8 8 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  13. Overview An Example Comparison to Actual Solution Conclusion Use Finite Differences with 8 Intervals to Solve the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e h : = 1 − 0 = 1 0 + n 1 : = x n 8 , 8 8 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  14. Overview An Example Comparison to Actual Solution Conclusion Use Finite Differences with 8 Intervals to Solve the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e h : = 1 − 0 = 1 0 + n 1 : = x n 8 , 8 8 y ′′ − 2 xy ′ − 2 y = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  15. Overview An Example Comparison to Actual Solution Conclusion Use Finite Differences with 8 Intervals to Solve the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e h : = 1 − 0 = 1 0 + n 1 : = x n 8 , 8 8 y ′′ − 2 xy ′ − 2 y = 0 y ( x + h ) − 2 y ( x )+ y ( x − h ) h 2 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  16. Overview An Example Comparison to Actual Solution Conclusion Use Finite Differences with 8 Intervals to Solve the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e h : = 1 − 0 = 1 0 + n 1 : = x n 8 , 8 8 y ′′ − 2 xy ′ − 2 y = 0 y ( x + h ) − 2 y ( x )+ y ( x − h ) − 2 xy ( x + h ) − y ( x − h ) h 2 2 h logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  17. Overview An Example Comparison to Actual Solution Conclusion Use Finite Differences with 8 Intervals to Solve the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e h : = 1 − 0 = 1 0 + n 1 : = x n 8 , 8 8 y ′′ − 2 xy ′ − 2 y = 0 y ( x + h ) − 2 y ( x )+ y ( x − h ) − 2 xy ( x + h ) − y ( x − h ) − 2 y ( x ) h 2 2 h logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  18. Overview An Example Comparison to Actual Solution Conclusion Use Finite Differences with 8 Intervals to Solve the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e h : = 1 − 0 = 1 0 + n 1 : = x n 8 , 8 8 y ′′ − 2 xy ′ − 2 y = 0 y ( x + h ) − 2 y ( x )+ y ( x − h ) − 2 xy ( x + h ) − y ( x − h ) − 2 y ( x ) = 0 h 2 2 h logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  19. Overview An Example Comparison to Actual Solution Conclusion Use Finite Differences with 8 Intervals to Solve the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e h : = 1 − 0 = 1 0 + n 1 : = x n 8 , 8 8 y ′′ − 2 xy ′ − 2 y = 0 y ( x + h ) − 2 y ( x )+ y ( x − h ) − 2 xy ( x + h ) − y ( x − h ) − 2 y ( x ) = 0 h 2 2 h − 2 h 2 y ( x ) � � y ( x + h ) − 2 y ( x )+ y ( x − h ) − xh y ( x + h ) − y ( x − h ) = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  20. Overview An Example Comparison to Actual Solution Conclusion Use Finite Differences with 8 Intervals to Solve the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e h : = 1 − 0 = 1 0 + n 1 : = x n 8 , 8 8 y ′′ − 2 xy ′ − 2 y = 0 y ( x + h ) − 2 y ( x )+ y ( x − h ) − 2 xy ( x + h ) − y ( x − h ) − 2 y ( x ) = 0 h 2 2 h − 2 h 2 y ( x ) � � y ( x + h ) − 2 y ( x )+ y ( x − h ) − xh y ( x + h ) − y ( x − h ) = 0 y ( x + h ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  21. Overview An Example Comparison to Actual Solution Conclusion Use Finite Differences with 8 Intervals to Solve the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e h : = 1 − 0 = 1 0 + n 1 : = x n 8 , 8 8 y ′′ − 2 xy ′ − 2 y = 0 y ( x + h ) − 2 y ( x )+ y ( x − h ) − 2 xy ( x + h ) − y ( x − h ) − 2 y ( x ) = 0 h 2 2 h − 2 h 2 y ( x ) � � y ( x + h ) − 2 y ( x )+ y ( x − h ) − xh y ( x + h ) − y ( x − h ) = 0 y ( x + h )( 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  22. Overview An Example Comparison to Actual Solution Conclusion Use Finite Differences with 8 Intervals to Solve the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e h : = 1 − 0 = 1 0 + n 1 : = x n 8 , 8 8 y ′′ − 2 xy ′ − 2 y = 0 y ( x + h ) − 2 y ( x )+ y ( x − h ) − 2 xy ( x + h ) − y ( x − h ) − 2 y ( x ) = 0 h 2 2 h − 2 h 2 y ( x ) � � y ( x + h ) − 2 y ( x )+ y ( x − h ) − xh y ( x + h ) − y ( x − h ) = 0 y ( x + h )( 1 − xh ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  23. Overview An Example Comparison to Actual Solution Conclusion Use Finite Differences with 8 Intervals to Solve the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e h : = 1 − 0 = 1 0 + n 1 : = x n 8 , 8 8 y ′′ − 2 xy ′ − 2 y = 0 y ( x + h ) − 2 y ( x )+ y ( x − h ) − 2 xy ( x + h ) − y ( x − h ) − 2 y ( x ) = 0 h 2 2 h − 2 h 2 y ( x ) � � y ( x + h ) − 2 y ( x )+ y ( x − h ) − xh y ( x + h ) − y ( x − h ) = 0 y ( x + h )( 1 − xh ) + y ( x ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  24. Overview An Example Comparison to Actual Solution Conclusion Use Finite Differences with 8 Intervals to Solve the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e h : = 1 − 0 = 1 0 + n 1 : = x n 8 , 8 8 y ′′ − 2 xy ′ − 2 y = 0 y ( x + h ) − 2 y ( x )+ y ( x − h ) − 2 xy ( x + h ) − y ( x − h ) − 2 y ( x ) = 0 h 2 2 h − 2 h 2 y ( x ) � � y ( x + h ) − 2 y ( x )+ y ( x − h ) − xh y ( x + h ) − y ( x − h ) = 0 � y ( x + h )( 1 − xh ) + y ( x ) − 2 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  25. Overview An Example Comparison to Actual Solution Conclusion Use Finite Differences with 8 Intervals to Solve the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e h : = 1 − 0 = 1 0 + n 1 : = x n 8 , 8 8 y ′′ − 2 xy ′ − 2 y = 0 y ( x + h ) − 2 y ( x )+ y ( x − h ) − 2 xy ( x + h ) − y ( x − h ) − 2 y ( x ) = 0 h 2 2 h − 2 h 2 y ( x ) � � y ( x + h ) − 2 y ( x )+ y ( x − h ) − xh y ( x + h ) − y ( x − h ) = 0 � − 2 − 2 h 2 � y ( x + h )( 1 − xh ) + y ( x ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  26. Overview An Example Comparison to Actual Solution Conclusion Use Finite Differences with 8 Intervals to Solve the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e h : = 1 − 0 = 1 0 + n 1 : = x n 8 , 8 8 y ′′ − 2 xy ′ − 2 y = 0 y ( x + h ) − 2 y ( x )+ y ( x − h ) − 2 xy ( x + h ) − y ( x − h ) − 2 y ( x ) = 0 h 2 2 h − 2 h 2 y ( x ) � � y ( x + h ) − 2 y ( x )+ y ( x − h ) − xh y ( x + h ) − y ( x − h ) = 0 � − 2 − 2 h 2 � y ( x + h )( 1 − xh ) + y ( x ) + y ( x − h ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  27. Overview An Example Comparison to Actual Solution Conclusion Use Finite Differences with 8 Intervals to Solve the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e h : = 1 − 0 = 1 0 + n 1 : = x n 8 , 8 8 y ′′ − 2 xy ′ − 2 y = 0 y ( x + h ) − 2 y ( x )+ y ( x − h ) − 2 xy ( x + h ) − y ( x − h ) − 2 y ( x ) = 0 h 2 2 h − 2 h 2 y ( x ) � � y ( x + h ) − 2 y ( x )+ y ( x − h ) − xh y ( x + h ) − y ( x − h ) = 0 � − 2 − 2 h 2 � y ( x + h )( 1 − xh ) + y ( x ) + y ( x − h )( 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  28. Overview An Example Comparison to Actual Solution Conclusion Use Finite Differences with 8 Intervals to Solve the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e h : = 1 − 0 = 1 0 + n 1 : = x n 8 , 8 8 y ′′ − 2 xy ′ − 2 y = 0 y ( x + h ) − 2 y ( x )+ y ( x − h ) − 2 xy ( x + h ) − y ( x − h ) − 2 y ( x ) = 0 h 2 2 h − 2 h 2 y ( x ) � � y ( x + h ) − 2 y ( x )+ y ( x − h ) − xh y ( x + h ) − y ( x − h ) = 0 � − 2 − 2 h 2 � y ( x + h )( 1 − xh ) + y ( x ) + y ( x − h )( 1 + xh ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  29. Overview An Example Comparison to Actual Solution Conclusion Use Finite Differences with 8 Intervals to Solve the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e h : = 1 − 0 = 1 0 + n 1 : = x n 8 , 8 8 y ′′ − 2 xy ′ − 2 y = 0 y ( x + h ) − 2 y ( x )+ y ( x − h ) − 2 xy ( x + h ) − y ( x − h ) − 2 y ( x ) = 0 h 2 2 h − 2 h 2 y ( x ) � � y ( x + h ) − 2 y ( x )+ y ( x − h ) − xh y ( x + h ) − y ( x − h ) = 0 � − 2 − 2 h 2 � y ( x + h )( 1 − xh ) + y ( x ) + y ( x − h )( 1 + xh ) = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  30. Overview An Example Comparison to Actual Solution Conclusion Use Finite Differences with 8 Intervals to Solve the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e h : = 1 − 0 = 1 0 + n 1 : = x n 8 , 8 8 y ′′ − 2 xy ′ − 2 y = 0 y ( x + h ) − 2 y ( x )+ y ( x − h ) − 2 xy ( x + h ) − y ( x − h ) − 2 y ( x ) = 0 h 2 2 h − 2 h 2 y ( x ) � � y ( x + h ) − 2 y ( x )+ y ( x − h ) − xh y ( x + h ) − y ( x − h ) = 0 � − 2 − 2 h 2 � y ( x + h )( 1 − xh ) + y ( x ) + y ( x − h )( 1 + xh ) = 0 − 2 − 2 h 2 � � y ( x n + h )( 1 − x n h )+ y ( x n ) + y ( x n − h )( 1 + x n h ) = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  31. Overview An Example Comparison to Actual Solution Conclusion Use Finite Differences with 8 Intervals to Solve the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e h : = 1 − 0 = 1 0 + n 1 : = x n 8 , 8 8 y ′′ − 2 xy ′ − 2 y = 0 y ( x + h ) − 2 y ( x )+ y ( x − h ) − 2 xy ( x + h ) − y ( x − h ) − 2 y ( x ) = 0 h 2 2 h − 2 h 2 y ( x ) � � y ( x + h ) − 2 y ( x )+ y ( x − h ) − xh y ( x + h ) − y ( x − h ) = 0 � − 2 − 2 h 2 � y ( x + h )( 1 − xh ) + y ( x ) + y ( x − h )( 1 + xh ) = 0 − 2 − 2 h 2 � � y ( x n + h )( 1 − x n h )+ y ( x n ) + y ( x n − h )( 1 + x n h ) = 0 � � � � � � 1 − 1 − 2 − 1 1 + 1 y n + 1 64 n + y n + y n − 1 64 n = 0 32 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  32. Overview An Example Comparison to Actual Solution Conclusion Use Finite Differences with 8 Intervals to Solve the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  33. Overview An Example Comparison to Actual Solution Conclusion Use Finite Differences with 8 Intervals to Solve the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e Boundaries : logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  34. Overview An Example Comparison to Actual Solution Conclusion Use Finite Differences with 8 Intervals to Solve the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e � � � � � � 1 − 1 − 2 − 1 1 + 1 Boundaries : y n + 1 + y n + y n − 1 = 0 64 n 64 n 32 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  35. Overview An Example Comparison to Actual Solution Conclusion Use Finite Differences with 8 Intervals to Solve the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e � � � � � � 1 − 1 − 2 − 1 1 + 1 Boundaries : y n + 1 + y n + y n − 1 = 0 64 n 64 n 32 � � 63 − 65 65 n = 1 : 64 + y 1 + y 0 = y 2 0 32 64 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  36. Overview An Example Comparison to Actual Solution Conclusion Use Finite Differences with 8 Intervals to Solve the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e � � � � � � 1 − 1 − 2 − 1 1 + 1 Boundaries : y n + 1 + y n + y n − 1 = 0 64 n 64 n 32 � � 63 − 65 65 n = 1 : 64 + y 1 + y 0 = y 2 0 32 64 � � 63 − 65 − 65 64 + y 1 = y 2 32 64 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  37. Overview An Example Comparison to Actual Solution Conclusion Use Finite Differences with 8 Intervals to Solve the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e � � � � � � 1 − 1 − 2 − 1 1 + 1 Boundaries : y n + 1 + y n + y n − 1 = 0 64 n 64 n 32 � � 63 − 65 65 n = 1 : 64 + y 1 + y 0 = y 2 0 32 64 � � 63 − 65 − 65 64 + y 1 = y 2 32 64 � � 57 − 65 71 n = 7 : 64 + y 7 + y 6 = y 8 0 32 64 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  38. Overview An Example Comparison to Actual Solution Conclusion Use Finite Differences with 8 Intervals to Solve the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e � � � � � � 1 − 1 − 2 − 1 1 + 1 Boundaries : y n + 1 + y n + y n − 1 = 0 64 n 64 n 32 � � 63 − 65 65 n = 1 : 64 + y 1 + y 0 = y 2 0 32 64 � � 63 − 65 − 65 64 + y 1 = y 2 32 64 � � 57 − 65 71 n = 7 : 64 + y 7 + y 6 = y 8 0 32 64 � � − 65 71 − e 57 + y 6 = y 7 32 64 64 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  39. Overview An Example Comparison to Actual Solution Conclusion Use Finite Differences with 8 Intervals to Solve the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  40. Overview An Example Comparison to Actual Solution Conclusion Use Finite Differences with 8 Intervals to Solve the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  41. Overview An Example Comparison to Actual Solution Conclusion Use Finite Differences with 8 Intervals to Solve the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  42. Overview An Example Comparison to Actual Solution Conclusion Use Finite Differences with 8 Intervals to Solve the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  43. Overview An Example Comparison to Actual Solution Conclusion Use Finite Differences with 8 Intervals to Solve the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  44. Overview An Example Comparison to Actual Solution Conclusion Use Finite Differences with 8 Intervals to Solve the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  45. Overview An Example Comparison to Actual Solution Conclusion Actual Solution of the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  46. Overview An Example Comparison to Actual Solution Conclusion Actual Solution of the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e e x 2 y ( x ) = logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  47. Overview An Example Comparison to Actual Solution Conclusion Actual Solution of the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e e x 2 y ( x ) = y ( 0 ) = 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  48. Overview An Example Comparison to Actual Solution Conclusion Actual Solution of the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e e x 2 y ( x ) = √ y ( 0 ) = 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  49. Overview An Example Comparison to Actual Solution Conclusion Actual Solution of the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e e x 2 y ( x ) = √ y ( 0 ) = 1 y ( 1 ) = e logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  50. Overview An Example Comparison to Actual Solution Conclusion Actual Solution of the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e e x 2 y ( x ) = √ √ y ( 0 ) = 1 y ( 1 ) = e logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  51. Overview An Example Comparison to Actual Solution Conclusion Actual Solution of the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e e x 2 y ( x ) = √ √ y ( 0 ) = 1 y ( 1 ) = e 2 xe x 2 y ′ ( x ) = logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  52. Overview An Example Comparison to Actual Solution Conclusion Actual Solution of the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e e x 2 y ( x ) = √ √ y ( 0 ) = 1 y ( 1 ) = e 2 xe x 2 y ′ ( x ) = 2 e x 2 + 4 x 2 e x 2 y ′′ ( x ) = logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  53. Overview An Example Comparison to Actual Solution Conclusion Actual Solution of the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e e x 2 y ( x ) = √ √ y ( 0 ) = 1 y ( 1 ) = e 2 xe x 2 y ′ ( x ) = 2 e x 2 + 4 x 2 e x 2 y ′′ ( x ) = 2 y + 2 xy ′ = logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  54. Overview An Example Comparison to Actual Solution Conclusion Actual Solution of the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e e x 2 y ( x ) = √ √ y ( 0 ) = 1 y ( 1 ) = e 2 xe x 2 y ′ ( x ) = 2 e x 2 + 4 x 2 e x 2 y ′′ ( x ) = √ 2 y + 2 xy ′ = logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  55. Overview An Example Comparison to Actual Solution Conclusion Actual Solution of the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  56. Overview An Example Comparison to Actual Solution Conclusion Actual Solution of the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  57. Overview An Example Comparison to Actual Solution Conclusion Actual Solution of the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  58. Overview An Example Comparison to Actual Solution Conclusion Actual Solution of the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  59. Overview An Example Comparison to Actual Solution Conclusion Actual Solution of the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  60. Overview An Example Comparison to Actual Solution Conclusion Actual Solution of the Boundary Value Problem y ′′ − 2 xy ′ − 2 y = 0, y ( 0 ) = 1, y ( 1 ) = e logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  61. Overview An Example Comparison to Actual Solution Conclusion The Finite Difference Method logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  62. Overview An Example Comparison to Actual Solution Conclusion The Finite Difference Method 1. The finite difference method is usually applied to partial differential equations logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  63. Overview An Example Comparison to Actual Solution Conclusion The Finite Difference Method 1. The finite difference method is usually applied to partial differential equations (hence the term “grid”). logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  64. Overview An Example Comparison to Actual Solution Conclusion The Finite Difference Method 1. The finite difference method is usually applied to partial differential equations (hence the term “grid”). 2. The solution is typically not known. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

  65. Overview An Example Comparison to Actual Solution Conclusion The Finite Difference Method 1. The finite difference method is usually applied to partial differential equations (hence the term “grid”). 2. The solution is typically not known. 3. One way to get an indication that we are close to a solution is to refine the grid and compare consecutive approximations. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Finite Difference Method

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