The Legendre multiwavelet Galerkin method is adopted to give the approximate solution for the nonlinear fractional partial differential equations (NFPDEs). The Legendre multiwavelet properties are presented. The main characteristic of this approach is using these properties together with the Galerkin method to reduce the NFPDEs to the solution of nonlinear system of algebraic equations. We presented the numerical results and a comparison with the exact solution in the cases when we have an exact solution to demonstrate the applicability and efficiency of the method. The fractional derivative is described in the Caputo sense.
Nowadays, fractional differential equations have garnered a great deal of attention and appreciation recently due to its ability to provide an accurate description of different nonlinear phenomena. The process of development of models based on fractional order differential systems has lately gained popularity in the investigation of dynamical systems. The advantage of fractional order systems is that they allow greater degrees of freedom in the model. The field of chaos has also snatched the attention of the researchers and this contributes to a large amount of the current research these days.
In recent decades, fractional calculus has found diverse applications in different scientific and technological fields [
The aim of this paper is to expand the application of Legendre multiwavelet Galerkin method to provide approximate solutions for initial value problems of fractional nonlinear partial differential equations and to make comparison with that obtained by other numerical methods.
In this section, we give the definition of the RiemannLiouville fractional derivative and fractional integral with some basic properties.
The left sided RiemannLiouville fractional integral of order
The (left sided) Caputo fractional derivative of
Wavelets constitute a family of functions constructed from dilation and translation of a single function called the mother wavelet. When the dilation parameter
Legendre multiwavelets
A function
Consider the nonlinear fractional partial differential equation
Let
A Galerkin approximation to (
The expansion coefficients
To demonstrate the effectiveness of the method, here we consider some linear fractional partial differential equations. The Legendre wavelets are defined only for
Consider the nonlinear timefractional diffusion equation in absence of both external force and reaction term [
We applied the method presented in this paper for
Approximate solutions for Example
















 
0.0  0.25  0.25  0.25  0.25  0.25  0.25  0.25  0.25  0.25  0.25 
0.50  0.50  0.50  0.50  0.50  0.50  0.50  0.50  0.50  0.50  
0.75  0.75  0.75  0.75  0.75  0.75  0.75  0.75  0.75  0.75  
1.00  1.00  1.00  1.00  1.00  1.00  1.00  1.00  1.00  1.00  


0.25  0.25  0.95546  0.93263  0.81419  0.78225  0.68960  0.66479  0.50  0.50  0.50 
0.50  1.20546  1.18263  1.06419  1.03225  0.93960  0.91479  0.75  0.75  0.75  
0.75  1.45546  1.43263  1.31419  1.28225  1.18960  1.16479  1.00  1.00  1.00  
1.00  1.70546  1.68263  1.56419  1.53225  1.43960  1.41479  1.25  1.25  1.25  


0.50  0.25  1.13882  1.30877  1.04788  1.13276  0.94783  0.98312  0.75  0.75  0.75 
0.50  1.38882  1.55877  1.29788  1.38276  1.19783  1.23312  1.00  1.00  1.00  
0.75  1.63882  1.80877  1.54788  1.63276  1.44783  1.48312  1.25  1.25  1.25  
1.00  1.88882  2.05877  1.79788  1.88276  1.69783  1.73312  1.50  1.50  1.50  


0.75  0.25  1.26745  1.37843  1.22721  1.30152  1.16441  1.20499  1.00  1.00  1.00 
0.50  1.51745  1.62843  1.47721  1.55152  1.41441  1.45499  1.25  1.25  1.25  
0.75  1.76745  1.87843  1.72721  1.80152  1.66441  1.70499  1.50  1.50  1.50  
1.00  2.01745  2.12843  1.97721  2.05152  1.91441  1.95499  1.75  1.75  1.75  


1.00  0.25  1.36985  1.14160  1.37838  1.28854  1.35773  1.33039  1.25  1.25  1.25 
0.50  1.61985  1.39160  1.62838  1.53854  1.60773  1.58039  1.50  1.50  1.50  
0.75  1.86985  1.64160  1.87838  1.78854  1.85773  1.83039  1.75  1.75  1.75  
1.00  2.11985  1.89160  2.12838  2.03854  2.10773  2.08039  2.00  2.00  2.00 
(a) Plot of
Plot of
Consider the fractional nonlinear KleinGordon equation [
We applied the method presented in this paper for
Approximate solutions for Example










 
0.25  0.25  0.918665978  1.132388298  1.164570723  1.188858256 
0.50  1.023903100  1.316468982  1.361000156  1.394880289  
0.75  1.099331862  1.470741307  1.527621230  1.571093962  
1.00  1.132379628  1.582632636  1.651861308  1.704926639  


0.50  0.25  0.670529477  0.862842686  0.954935119  1.028617931 
0.50  0.683586761  0.936088923  1.063084555  1.165329668  
0.75  0.666835687  0.979526801  1.141425631  1.272233047  
1.00  0.607703616  0.980583683  1.177385711  1.336755430  


0.75  0.25  0.502994454  0.552019181  0.676793302  0.7897781568 
0.50  0.458476525  0.501019780  0.671719555  0.8269013752  
0.75  0.384150235  0.420212023  0.636837450  0.8342162344  
1.00  0.267442949  0.297023267  0.559574347  0.799150097  


1.00  0.25  0.416060911  0.313169842  0.388441429  0.4954341090 
0.50  0.348572389  0.173995974  0.272945982  0.415723105  
0.75  0.251275507  0.005013748  0.127642174  0.306203742  
1.00  0.111597629  −0.206349475  −0.060042629  0.154303383 
(a) Plot of
Plot of
Consider the following nonlinear timefractional equation [
We applied the method presented in this paper for
Numerical values when








0.2  0.25  0.123635  0.104573  0.103750  0.112844  0.092812 
0.50  0.247270  0.209146  0.207499  0.225688  0.185624  
0.75  0.370905  0.313720  0.311249  0.311249  0.278436  
1.00  0.494540  0.418293  0.414999  0.451375  0.371248  


0.4  0.25  0.177148  0.177229  0.172012  0.164004  0.161241 
0.50  0.354295  0.354458  0.344025  0.328008  0.322483  
0.75  0.531443  0.531686  0.516037  0.492011  0.483725  
1.00  0.708590  0.708915  0.688050  0.656015  0.644967  


0.6  0.25  0.226965  0.230500  0.215641  0.243862  0.205288 
0.50  0.453931  0.461000  0.431283  0.487721  0.410577  
0.75  0.680896  0.691499  0.646924  0.731581  0.615866  
1.00  0.907861  0.921999  0.862566  0.975441  0.821155 
Numerical values when








0.2  0.25  0.080835  0.078306  0.077933  0.078787  0.070083 
0.50  0.161671  0.156612  0.155865  0.157574  0.140166  
0.75  0.242596  0.234919  0.233798  0.236361  0.210249  
1.00  0.323342  0.313225  0.311730  0.315148  0.280332  


0.4  0.25  0.135446  0.136806  0.134855  0.128941  0.129648 
0.50  0.270892  0.273612  0.269710  0.257881  0.259297  
0.75  0.406338  0.410418  0.404565  0.386821  0.388945  
1.00  0.541784  0.547225  0.539420  0.515762  0.518594  


0.6  0.25  0.185529  0.185146  0.179990  0.177238  0.178696 
0.50  0.371057  0.370292  0.359979  0.354477  0.357393  
0.75  0.556586  0.555437  0.539969  0.531715  0.536089  
1.00  0.742114  0.740583  0.719958  0.7089541  0.714786 
Numerical values when









0.2  0.25  0.05  0.049989  0.050309  0.050000  0.05  0.05 
0.50  0.10  0.099978  0.100619  0.100000  0.10  0.10  
0.75  0.15  0.149968  0.150928  0.150001  0.15  0.15  
1.00  0.20  0.199957  0.201237  0.200001  0.20  0.20  


0.4  0.25  0.10  0.099645  0.101894  0.100023  0.10  0.10 
0.50  0.20  0.199290  0.203787  0.200046  0.20  0.20  
0.75  0.30  0.298935  0.305681  0.300069  0.30  0.30  
1.00  0.40  0.398580  0.407575  0.400092  0.40  0.40  


0.6  0.25  0.15  0.147158  0.153094  0.150411  0.15  0.15 
0.50  0.30  0.294317  0.306188  0.300823  0.30  0.30  
0.75  0.45  0.441475  0.459282  0.451234  0.45  0.45  
1.00  0.60  0.588634  0.612376  0.601646  0.60  0.60 
Plot of
Plot of
Consider the following time fractional advection nonhomogeneous equation [
We applied the method presented in this paper for
Substituting (
Numerical values when













 
0.2  0.25  0.243843  0.216416  0.153820  0.150051  0.09  0.09  0.09 
0.50  0.351274  0.309228  0.232395  0.220134  0.14  0.14  0.14  
0.75  0.458705  0.402040  0.310971  0.290218  0.19  0.19  0.19  
1.00  0.566136  0.494853  0.389547  0.360301  0.24  0.24  0.24  


0.4  0.25  0.534552  0.474657  0.388100  0.363967  0.26  0.26  0.26 
0.50  0.667613  0.635899  0.514376  0.493616  0.36  0.36  0.36  
0.75  0.800673  0.797140  0.640652  0.623264  0.46  0.46  0.46  
1.00  0.933734  0.958382  0.766928  0.752913  0.56  0.56  0.56  


0.6  0.25  0.934713  0.774722  0.726195  0.641747  0.51  0.51  0.51 
0.50  1.083891  0.980011  0.890481  0.820443  0.66  0.66  0.66  
0.75  1.233068  1.185300  1.054767  0.999140  0.81  0.81  0.81  
1.00  1.382245  1.390589  1.219052  1.177837  0.96  0.96  0.96 
Plot of
Plot of
In this study, it is shown how Legendre multiwavelet can be applied to provide approximate solutions for initial value problems of fractional nonlinear partial differential equations. The Legendre multiwavelet properties are presented. The main characteristic of this approach is using these properties together with the Galerkin method to reduce the NFPDEs to the solution of nonlinear system of algebraic equations. In addition, we compered our results with that obtained by other numerical methods. The results show that the Legendre multiwavelet is a powerful mathematical tool for fractional nonlinear partial differential equations. We used Mathematica and Maple programs for computations in this paper.
The authors declare that there is no conflict of interests regarding the publication of this paper.