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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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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