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Initial value problem for fractional Volterra integro-differential equations with Caputo derivative

Dedicated to Tomás Caraballo on his 60th birthday.

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  • In this paper, we consider the time-fractional Volterra integro-differential equations with Caputo derivative. For globally Lispchitz source term, we investigate the global existence for a mild solution. The main tool is to apply the Banach fixed point theorem on some new weighted spaces combining some techniques on the Wright functions. For the locally Lipschitz case, we study the existence of local mild solutions to the problem and provide a blow-up alternative for mild solutions. We also establish the problem of continuous dependence with respect to initial data. Finally, we present some examples to illustrate the theoretical results.

    Mathematics Subject Classification: 35K55, 35K70, 35K92, 47A52, 47J06.

    Citation:

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  • Figure 1.  Ex1. The solutions $ u(x,t) $ at $ t \in \{0.1, 0.5, 0.9\} $ for $ \alpha = 0.2 $ and $ \epsilon \in \{0.1, 0.01, 0.001\} $

    Figure 2.  Ex1. The solutions $ u(x,t) $ on $ (x,t) \in (0,\pi) \times (0,1) $ for $ \alpha = 0.2 $ and $ \epsilon \in \{0.1, 0.01, 0.001\} $

    Figure 3.  Ex2. The solutions $ u(x,t) $ at $ t \in \{0.1, 0.5, 0.9\} $ for $ \alpha = 0.9 $ and $ \epsilon \in \{0.1, 0.01, 0.001\} $

    Figure 4.  Ex2. The solutions $ u(x,t) $ on $ (x,t) \in (0,\pi) \times (0,1) $ for $ \alpha = 0.9 $ and $ \epsilon \in \{0.1, 0.01, 0.001\} $

    Table 1.  Ex1. The error estimation for $ \alpha = 0.2 $ and $ t \in \{0.1, 0.5, 0.9\} $

    $ \epsilon = |\alpha^* - \alpha| $ $ N(j) = 10,\; P =10,\; M = N =50 $
    Calculative error Percent error $ \delta^{ \alpha'}_ \alpha $
    $ \mathrm{Error}^{ \alpha^*}_ \alpha $(0.1) 0.1 0.011036875514009 15.46 %
    0.01 0.004874547920700 6.83 %
    0.001 0.002176754401323 3.05 %
    $ \mathrm{Error}^{ \alpha^*}_ \alpha $(0.5) 0.1 0.051353021961229 14.38 %
    0.01 0.027169280169359 7.61 %
    0.001 0.012084128051511 3.38 %
    $ \mathrm{Error}^{ \alpha^*}_ \alpha $(0.9) 0.1 0.074877070037197 11.65 %
    0.01 0.042748723217464 6.65 %
    0.001 0.028729381706881 4.47 %
     | Show Table
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    Table 2.  Ex2. The error estimation for $ \alpha = 0.9 $ and $ t \in \{0.1, 0.5, 0.9\} $

    $ \epsilon = |\alpha^* - \alpha| $ $ N(j) = 10,\; P =10,\; M = N =50 $
    Calculative error Percent error $ \delta^{ \alpha'}_ \alpha $
    $ \mathrm{Error}^{ \alpha^*}_ \alpha $(0.1) 0.1 0.601407040658915 83.81 %
    0.01 0.143648723675101 20.02 %
    0.001 0.027371563705550 3.81 %
    $ \mathrm{Error}^{ \alpha^*}_ \alpha $(0.5) 0.1 0.643179867112674 79.89 %
    0.01 0.167904683423887 20.85 %
    0.001 0.025589340748129 3.18 %
    $ \mathrm{Error}^{ \alpha^*}_ \alpha $(0.9) 0.1 0.629790962460797 61.54 %
    0.01 0.219601251416672 21.46 %
    0.001 0.040291675260527 3.94 %
     | Show Table
    DownLoad: CSV
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