IBVPInitial Boundary Value Problem
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Notice that for the IBVP (2.1)-(2.3) all the above mentioned types of solutions can be represented by one and the same D'Alembert-type formula so we could omit their definitions.
Now we present numerical methods to solve the IBVP (2.1)-(2.3) which error we analyze below.
As in [34], we consider the linear parabolic IBVP: find u : [bar.Q] [right arrow] R such that
In particular, the term [[absolute value of [[partial derivative].sub.t].sup.[rho]][[partial derivative].sup.2.sub.t]u with [rho] > 0, by Boltzmann's superposition principle, indicates that the wave speed depends on the velocity of the wave; [[integral].sup.t.sub.0]g(t - [tau])[DELTA]u(x, [tau])d[tau], the so-called viscoelastic damping, is incorporated in IBVP (1) to characterize the hereditary properties.
The proposed method could be briefly described by considering simple IBVP, (EQUATION), with initial condition i.e.
It is called the first IBVP for nonlinear time fractional diffusion equation.
with [alpha] = 1 - [p.sup.2.sub.1], reduce the above IBVP (6)-(7) to BVP of ODE as follows:
In the present paper we study the existence, uniqueness and other properties of the solutions of IBVP (1.4)-(1.5) under some suitable conditions on the functions involved in (1.4), (1.5).
For this purpose and for simplicity, we solve the IBVP (16) in Lagrangian coordinates.
In order to apply the finite volume technique, the IBVP has to be transformed into pure divergence form.
The similarity variables of the symmetry that leaves the whole IBVP invariant will lead to the reduction of IBVP of PDE (5) to a BVP of ODE of the form