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Modified Wavelet Full-Approximation Scheme for the Numerical Solution of Nonlinear Volterra integral and integro-differential Equations

S. C. Shiralashetti
S. C. Shiralashetti
 y 
R. A. Mundewadi
R. A. Mundewadi
   | 16 nov 2016
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Publicado en línea: 16 nov 2016
Páginas: 529 - 546
Recibido: 07 feb 2016
Aceptado: 16 nov 2016
DOI: https://doi.org/10.21042/AMNS.2016.2.00042
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© 2016 S. C. Shiralashetti, R. A. Mundewadi, published by Sciendo
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Fig. 1

Comparison of numerical solutions with exact solution of test problem 5.1, for N=64.
Comparison of numerical solutions with exact solution of test problem 5.1, for N=64.

Fig. 2

Comparison of numerical solutions with exact solution of test problem 5.2, for N=64.
Comparison of numerical solutions with exact solution of test problem 5.2, for N=64.

Fig. 3

Comparison of numerical solutions with exact solution of test problem 5.3, for N=64.
Comparison of numerical solutions with exact solution of test problem 5.3, for N=64.

Fig. 4

Comparison of numerical solutions with exact solution of test problem 5.4, for N=64.
Comparison of numerical solutions with exact solution of test problem 5.4, for N=64.

Fig. 5

Comparison of numerical solutions with exact solution of test problem 5.5, for N=64.
Comparison of numerical solutions with exact solution of test problem 5.5, for N=64.

Numerical results of test problem 5.1, for N = 8.

tExactFASWFASMWFAS
00000
0.14280.12240.12240.12240.1224
0.28570.20400.20430.20430.2043
0.42850.24480.24570.24570.2457
0.57140.24480.24630.24630.2463
0.71420.20400.20550.20550.2055
0.85710.12240.12310.12310.1231
100.00010.00010.0001

Numerical results of test problem 5.2, for N = 16.

tExactFASWFASMWFAS
0 0 0 0 0
0.0666 0.0666 0.0667 0.0667 0.0667
0.1333 0.1329 0.1332 0.1332 0.1332
0.2000 0.1986 0.1994 0.1994 0.1994
0.2666 0.2635 0.2648 0.2648 0.2648
0.3333 0.3271 0.3291 0.3291 0.3291
0.4000 0.3894 0.3922 0.3922 0.3922
0.4666 0.4499 0.4537 0.4537 0.4537
0.5333 0.5084 0.5133 0.5130 0.5130
0.6000 0.5646 0.5708 0.5703 0.5703
0.6666 0.6183 0.6259 0.6253 0.6253
0.7333 0.6693 0.6784 0.6776 0.6776
0.8000 0.7173 0.7280 0.7270 0.7270
0.8666 0.7621 0.7746 0.7734 0.7734
0.9333 0.8036 0.8174 0.8164 0.8164
1 0.8414 0.8449 0.8441 0.8441

Maximum error and CPU time (in seconds) of the methods of test problem 5.5.

NMethodsEmaxSetup timeRunning timeTotal time
16FASWFASMWFAS2:81e − 042:81e − 042:81e − 040.00540.00390.00250.01250.01130.01080.01800.01520.0133
32FASWFASMWFAS6:56e − 056:56e − 056:56e − 050.01000.00300.00190.01690.01610.01580.02680.01900.0177
64FASWFASMWFAS1:57e − 051:57e − 051:57e − 050.02860.00300.00190.03680.03580.03530.06540.03880.0372
128FASWFASMWFAS3:69e − 063:69e − 063:69e − 060.10420.00330.00190.12400.11560.11670.22810.11890.1186

Maximum error and CPU time (in seconds) of the methods of test problem 5.3.

NMethodsEmaxSetup timeRunning timeTotal time
16FASWFASMWFAS3:57e − 023:57e − 023:57e − 020.00540.00310.00190.01210.01210.01080.01750.01520.0127
32FASWFASMWFAS1:66e − 021:66e − 021:66e − 020.00990.00310.00190.01720.01640.01560.02710.01950.0175
64FASWFASMWFAS8:06e − 038:06e − 038:06e − 030.02810.00310.00190.03700.03600.03570.06510.03900.0375
128FASWFASMWFAS3:96e − 033:96e − 033:96e − 030.10090.00440.00200.11680.13290.13000.21770.13730.1320

Maximum error and CPU time (in seconds) of the methods of test problem 5.4.

NMethodsEmaxSetup timeRunning timeTotal time
16FASWFASMWFAS5:72e − 045:60e − 045:60e − 040.00540.00300.00180.04040.03340.03150.04580.03650.0333
32FASWFASMWFAS2:81e − 042:77e − 042:76e − 040.01050.00300.00180.20840.20560.20570.21890.20860.2075
64FASWFASMWFAS1:38e − 041:37e − 041:37e − 040.03060.00300.00180.04840.04710.04560.07900.05020.0474
128FASWFASMWFAS6:87e − 056:85e − 056:85e − 050.11140.00440.00180.18100.17800.18040.29240.18230.1822

Maximum error and CPU time (in seconds) of the methods of test problem 5.2.

NMethodsEmaxSetup timeRunning timeTotal time
16FASWFASMWFAS1:38e − 021:28e − 021:28e − 020.06240.00480.00190.06360.04430.02560.12600.04910.0275
32FASWFASMWFAS7:08e − 036:80e − 036:80e − 030.00990.00310.00190.14570.14690.14750.15560.14990.1494
64FASWFASMWFAS3:56e − 033:49e − 033:49e − 030.24630.00310.00191.17611.13851.14561.42251.14161.1475
128FASWFASMWFAS1:79e − 031:77e − 031:77e − 030.18190.01250.001910.25919.53019.322010.44109.54269.3239

Maximum error and CPU time (in seconds) of the methods of test problem 5.1.

NMethodsEmaxSetup timeRunning timeTotal time
16 FASWFASMWFAS7:52e − 047:35e − 047:35e − 040.00540.00310.00190.03210.02640.02540.03750.02950.0273
32FASWFASMWFAS3:66e − 043:62e − 043:62e − 040.00990.00310.00190.14590.14790.14700.15580.15100.1489
64FASWFASMWFAS1:80e − 041:79e − 041:79e − 040.12610.00310.00190.15810.03690.03571.28420.04000.0376
128FASWFASMWFAS8:99e − 058:96e − 058:96e − 050.13070.00850.00190.21850.14190.11620.34920.15040.1182

Numerical results of test problem 5.3, for N = 16.

tExactFASWFASMWFAS
0 1 1.0357 1.0357 1.0357
0.0666 0.9977 0.9972 0.9972 0.9972
0.1333 0.9911 0.9901 0.9901 0.9901
0.2000 0.9800 0.9786 0.9786 0.9786
0.2666 0.9646 0.9628 0.9628 0.9628
0.3333 0.9449 0.9427 0.9427 0.9427
0.4000 0.9210 0.9185 0.9185 0.9185
0.4666 0.8930 0.8903 0.8903 0.8903
0.5333 0.8611 0.8582 0.8582 0.8582
0.6000 0.8253 0.8223 0.8223 0.8223
0.6666 0.7858 0.7829 0.7829 0.7829
0.7333 0.7429 0.7400 0.7400 0.7400
0.8000 0.6967 0.6939 0.6939 0.6939
0.8666 0.6473 0.6448 0.6448 0.6448
0.9333 0.5951 0.5929 0.5929 0.5929
1 0.5403 0.5383 0.5383 0.5383

Numerical results of test problem 5.5, for N = 32.

tExactFASWFASMWFAS
0 0 0 0 0
0.0322 -0.0322 -0.0322 -0.0322 -0.0322
0.0645 -0.0645 -0.0645 -0.0645 -0.0645
0.0967 -0.0967 -0.0967 -0.0967 -0.0967
0.1290 -0.1290 -0.1290 -0.1290 -0.1290
0.1612 -0.1612 -0.1612 -0.1612 -0.1612
0.1935 -0.1934 -0.1934 -0.1934 -0.1934
0.2258 -0.2255 -0.2255 -0.2255 -0.2255
0.2580 -0.2577 -0.2577 -0.2577 -0.2577
0.2903 -0.2897 -0.2897 -0.2897 -0.2897
0.3225 -0.3216 -0.3216 -0.3216 -0.3216
0.3548 -0.3535 -0.3535 -0.3535 -0.3535
0.3870 -0.3852 -0.3852 -0.3852 -0.3852
0.4193 -0.4167 -0.4168 -0.4168 -0.4168
0.4516 -0.4481 -0.4481 -0.4481 -0.4481
0.4838 -0.4793 -0.4793 -0.4793 -0.4793
0.5161 -0.5102 -0.5102 -0.5102 -0.5102
0.5483 -0.5409 -0.5409 -0.5409 -0.5409
0.5806 -0.5712 -0.5712 -0.5712 -0.5712
0.6129 -0.6012 -0.6013 -0.6013 -0.6013
0.6451 -0.6309 -0.6309 -0.6309 -0.6309
0.6774 -0.6601 -0.6601 -0.6601 -0.6601
0.7096 -0.6888 -0.6889 -0.6889 -0.6889
0.7419 -0.7171 -0.7172 -0.7172 -0.7172
0.7741 -0.7449 -0.7449 -0.7449 -0.7449
0.8064 -0.7720 -0.7721 -0.7721 -0.7721
0.8387 -0.7986 -0.7986 -0.7986 -0.7986
0.8709 -0.8244 -0.8245 -0.8245 -0.8245
0.9032 -0.8496 -0.8497 -0.8497 -0.8497
0.9354 -0.8740 -0.8741 -0.8741 -0.8741
0.9677 -0.8976 -0.8977 -0.8977 -0.8977
1 -0.9204 -0.9205 -0.9205 -0.9205

Numerical results of test problem 5.4, for N = 16.

tExactFASWFASMWFAS
0 0 0 0 0
0.0666 -0.0622 -0.0622 -0.0622 -0.0622
0.1333 -0.1155 -0.1155 -0.1155 -0.1155
0.2000 -0.1600 -0.1600 -0.1600 -0.1600
0.2666 -0.1955 -0.1957 -0.1957 -0.1957
0.3333 -0.2222 -0.2224 -0.2224 -0.2224
0.4000 -0.2400 -0.2403 -0.2403 -0.2403
0.4666 -0.2488 -0.2493 -0.2493 -0.2493
0.5333 -0.2488 -0.2494 -0.2494 -0.2494
0.6000 -0.2400 -0.2405 -0.2405 -0.2405
0.6666 -0.2222 -0.2227 -0.2227 -0.2227
0.7333 -0.1955 -0.1959 -0.1959 -0.1959
0.8000 -0.1600 -0.1602 -0.1602 -0.1602
0.8666 -0.1155 -0.1156 -0.1156 -0.1156
0.9333 -0.0622 -0.0622 -0.0622 -0.0622
1 0 0 0 0
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