Theoretical Prospects of the Solutions of Fractional Order Weakly
Singular Volterra Integro Differential Equations and their
Approximations with Convergence Analysis
01 Mar 2020
01 Mar 2020
22 Mar 2020
09 Jul 2020
10 Jul 2020
AbstractIn this work, we study a weakly singular Volterra integro differential
equation with Caputo type fractional derivative. First, we derive a
sufficient condition for the existence and uniqueness of the solution of
this problem based on the maximum norm. It is observed that the
condition depends on the domain of definition of the problem.
Thereafter, we show that this condition will be independent of the
domain of definition based on an equivalent weighted maximum norm. In
addition, we have also provided a procedure to extend the existence and
uniqueness of the solution in its domain of definition by partitioning
it. We also derive a sufficient condition under which the model problem
will provide an analytic solution. Next, we introduce a operator based
parameterized method to generate an approximate solution of this
problem. Convergence analysis of this approach is established here.
Next, we have optimized this solution based on least square method. For
this, residual minimization is used to obtain the optimal values of the
auxiliary parameter. In addition, we have also provided an error bound
based on this technique. Several numerical examples are produced to
clarify the effective behavior of the convergence of the present method.
Comparison of the standard method and optimized method based on residual
minimization signify the better accuracy of modified one. In addition,
we also consider an equivalent form of weakly singular integro
differential equation of a Heat transfer problem to show the
effectiveness of the present approach.