1 Introduction
2 Approximation of the Riesz-Caputo fractional derivative with variable order
2.1 Method I: Approximation based on piecewise constant interpolation
2.2 Method II: Approximation based on piecewise linear interpolation
2.3 Method III: Approximation based on piecewise quadratic interpolation
3 Numerical analysis: errors and experimental rates of convergence
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\(f\left( x\right) =\exp \left( x\right)\) and \(\alpha \left( x\right) =\exp \left( -x\right)\) (Table 1),
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\(f\left( x\right) =\exp \left( x\right)\) and \(\alpha \left( x\right) =\frac{1+x^2}{2}\) (Table 2),
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\(f\left( x\right) =\exp \left( x\right)\) and \(\alpha \left( x\right) =\sin \left( x\right) +1\) (Table 3).
x | \(\varDelta x\) | Method I | Method II | Method III | |||
---|---|---|---|---|---|---|---|
Error | Rate | Error | Rate | Error | Rate | ||
0.2 | 0.01 | 1.50965E-05 | – | 6.41933E-08 | – | 1.08056E-14 | – |
0.005 | 7.57269E-06 | 1.00 | 1.65880E-08 | 1.95 | 6.87771E-16 | 3.97 | |
0.0025 | 3.79284E-06 | 1.00 | 4.26599E-09 | 1.96 | 4.36702E-17 | 3.98 | |
0.00125 | 1.89813E-06 | 1.00 | 1.09273E-09 | 1.96 | 2.76712E-18 | 3.98 | |
0.4 | 0.01 | 2.42990E-05 | – | 7.68809E-08 | – | 9.25228E-15 | – |
0.005 | 1.21786E-05 | 1.00 | 1.94861E-08 | 1.98 | 5.77257E-16 | 4.00 | |
0.0025 | 6.09687E-06 | 1.00 | 4.92442E-09 | 1.98 | 3.60283E-17 | 4.00 | |
0.00125 | 3.05037E-06 | 1.00 | 1.24163E-09 | 1.99 | 2.24927E-18 | 4.00 | |
0.8 | 0.01 | 3.66332E-05 | – | 8.55870E-08 | – | 4.72786E-15 | – |
0.005 | 1.83466E-05 | 1.00 | 2.14589E-08 | 2.00 | 2.92187E-16 | 4.02 | |
0.0025 | 9.18088E-06 | 1.00 | 5.37533E-09 | 2.00 | 1.81207E-17 | 4.01 | |
0.00125 | 4.59236E-06 | 1.00 | 1.34564E-09 | 2.00 | 1.12653E-18 | 4.01 | |
1.2 | 0.01 | 4.88466E-05 | – | 9.92076E-08 | – | 1.41104E-15 | – |
0.005 | 2.44551E-05 | 1.00 | 2.48227E-08 | 2.00 | 8.62417E-17 | 4.03 | |
0.0025 | 1.22355E-05 | 1.00 | 6.20887E-09 | 2.00 | 5.31507E-18 | 4.02 | |
0.00125 | 6.11977E-06 | 1.00 | 1.55271E-09 | 2.00 | 3.29302E-19 | 4.01 | |
1.6 | 0.01 | 6.54281E-05 | – | 1.23339E-07 | – | 1.56315E-15 | – |
0.005 | 3.27508E-05 | 1.00 | 3.08444E-08 | 2.00 | 9.89008E-17 | 3.98 | |
0.0025 | 1.63846E-05 | 1.00 | 7.71249E-09 | 2.00 | 6.22459E-18 | 3.99 | |
0.00125 | 8.19461E-06 | 1.00 | 1.92832E-09 | 2.00 | 3.90593E-19 | 3.99 | |
1.8 | 0.01 | 7.64028E-05 | – | 1.40558E-07 | – | 3.16440E-15 | – |
0.005 | 3.82421E-05 | 1.00 | 3.51467E-08 | 2.00 | 1.98766E-16 | 3.99 | |
0.0025 | 1.91312E-05 | 1.00 | 8.78768E-09 | 2.00 | 1.24576E-17 | 4.00 | |
0.00125 | 9.56817E-06 | 1.00 | 2.19706E-09 | 2.00 | 7.79820E-19 | 4.00 |
x | \(\varDelta x\) | Method I | Method II | Method III | |||
---|---|---|---|---|---|---|---|
Error | Rate | Error | Rate | Error | Rate | ||
0.2 | 0.01 | 2.06366E-05 | – | 5.23628E-08 | – | 3.94876E-15 | – |
0.005 | 1.03373E-05 | 1.00 | 1.31554E-08 | 1.99 | 2.44466E-16 | 4.01 | |
0.0025 | 5.17346E-06 | 1.00 | 3.30045E-09 | 1.99 | 1.51746E-17 | 4.01 | |
0.00125 | 2.58795E-06 | 1.00 | 8.27193E-10 | 2.00 | 9.43733E-19 | 4.01 | |
0.4 | 0.01 | 2.52875E-05 | – | 6.95082E-08 | – | 6.46222E-15 | – |
0.005 | 1.26695E-05 | 1.00 | 1.75087E-08 | 1.99 | 4.00927E-16 | 4.01 | |
0.0025 | 6.34134E-06 | 1.00 | 4.40176E-09 | 1.99 | 2.49195E-17 | 4.01 | |
0.00125 | 3.17235E-06 | 1.00 | 1.10504E-09 | 1.99 | 1.55101E-18 | 4.01 | |
0.8 | 0.01 | 2.73888E-05 | – | 1.16803E-07 | – | 1.97114E-14 | – |
0.005 | 1.37389E-05 | 1.00 | 3.01898E-08 | 1.95 | 1.25493E-15 | 3.97 | |
0.0025 | 6.88124E-06 | 1.00 | 7.76567E-09 | 1.96 | 7.96998E-17 | 3.98 | |
0.00125 | 3.44373E-06 | 1.00 | 1.98958E-09 | 1.96 | 5.05113E-18 | 3.98 | |
1.2 | 0.01 | 4.47995E-05 | - | 8.55260E-08 | – | 6.66571E-16 | – |
0.005 | 2.24256E-05 | 1.00 | 2.13896E-08 | 2.00 | 4.26259E-17 | 3.97 | |
0.0025 | 1.12193E-05 | 1.00 | 5.34859E-09 | 2.00 | 2.69926E-18 | 3.98 | |
0.00125 | 5.61127E-06 | 1.00 | 1.33732E-09 | 2.00 | 1.69983E-19 | 3.99 | |
1.6 | 0.01 | 6.83960E-05 | – | 2.66922E-07 | – | 4.14854E-14 | – |
0.005 | 3.43000E-05 | 1.00 | 6.85320E-08 | 1.96 | 2.62248E-15 | 3.98 | |
0.0025 | 1.71769E-05 | 1.00 | 1.75197E-08 | 1.97 | 1.65496E-16 | 3.99 | |
0.00125 | 8.59546E-06 | 1.00 | 4.46293E-09 | 1.97 | 1.04289E-17 | 3.99 | |
1.8 | 0.01 | 7.20214E-05 | – | 1.28814E-07 | – | 4.46378E-15 | – |
0.005 | 3.60466E-05 | 1.00 | 3.22071E-08 | 2.00 | 2.79546E-16 | 4.00 | |
0.0025 | 1.80323E-05 | 1.00 | 8.05228E-09 | 2.00 | 1.74906E-17 | 4.00 | |
0.00125 | 9.01840E-06 | 1.00 | 2.01314E-09 | 2.00 | 1.09381E-18 | 4.00 |
x | \(\varDelta x\) | Method I | Method II | Method III | |||
---|---|---|---|---|---|---|---|
Error | Rate | Error | Rate | Error | Rate | ||
0.2 | 0.01 | 1.60723E-05 | – | 3.02310E-08 | 4.09406E-16 | ||
0.005 | 8.04511E-06 | 1.00 | 7.56003E-09 | 2.00 | 2.58751E-17 | 3.98 | |
0.0025 | 4.02481E-06 | 1.00 | 1.89034E-09 | 2.00 | 1.62750E-18 | 3.99 | |
0.00125 | 2.01297E-06 | 1.00 | 4.72631E-10 | 2.00 | 1.02088E-19 | 3.99 | |
0.4 | 0.01 | 2.36586E-05 | – | 5.19790E-08 | – | 2.00026E-15 | – |
0.005 | 1.18469E-05 | 1.00 | 1.30181E-08 | 2.00 | 1.23408E-16 | 4.02 | |
0.0025 | 5.92790E-06 | 1.00 | 3.25836E-09 | 2.00 | 7.64717E-18 | 4.01 | |
0.00125 | 2.96507E-06 | 1.00 | 8.15218E-10 | 2.00 | 4.75253E-19 | 4.01 | |
0.8 | 0.01 | 3.44676E-05 | – | 1.18646E-07 | – | 1.60320E-14 | – |
0.005 | 1.72791E-05 | 1.00 | 3.02127E-08 | 1.97 | 1.00487E-15 | 4.00 | |
0.0025 | 8.65136E-06 | 1.00 | 7.66646E-09 | 1.98 | 6.29514E-17 | 4.00 | |
0.00125 | 4.32873E-06 | 1.00 | 1.93990E-09 | 1.98 | 3.94203E-18 | 4.00 | |
1.2 | 0.01 | 1.94921E-05 | 1.10806E-07 | 2.29987E-14 | |||
0.005 | 9.78684E-06 | 0.99 | 2.93851E-08 | 1.91 | 1.50571E-15 | 3.93 | |
0.0025 | 4.90457E-06 | 1.00 | 7.74780E-09 | 1.92 | 9.81780E-17 | 3.94 | |
0.00125 | 2.45530E-06 | 1.00 | 2.03271E-09 | 1.93 | 6.37888E-18 | 3.94 | |
1.6 | 0.01 | 2.09175E-07 | – | 1.45639E-09 | – | 3.37524E-16 | – |
0.005 | 1.05104E-07 | 0.99 | 3.94470E-10 | 1.88 | 2.26121E-17 | 3.90 | |
0.0025 | 5.26960E-08 | 1.00 | 1.06208E-10 | 1.89 | 1.50804E-18 | 3.91 | |
0.00125 | 2.63879E-08 | 1.00 | 2.84493E-11 | 1.90 | 1.00174E-19 | 3.91 | |
1.8 | 0.01 | 1.48806E-05 | – | 9.56471E-08 | – | 2.12801E-14 | – |
0.005 | 7.47474E-06 | 0.99 | 2.56854E-08 | 1.90 | 1.41240E-15 | 3.91 | |
0.0025 | 3.74691E-06 | 1.00 | 6.85677E-09 | 1.91 | 9.33326E-17 | 3.92 | |
0.00125 | 1.87607E-06 | 1.00 | 1.82109E-09 | 1.91 | 6.14370E-18 | 3.93 |