Numerical Solutions of Fractional Integrodifferential Equations of Bratu Type by Using CAS Wavelets

A numerical method based on the CAS wavelets is presented for the fractional integrodifferential equations of Bratu type.The CAS wavelets operationalmatrix of fractional order integration is derived. A truncatedCASwavelets series together with this operational matrix is utilized to reduce the fractional integrodifferential equations to a system of algebraic equations.The solution of this system gives the approximation solution for the truncated limited 2(2M + 1). The convergence and error estimation of CAS wavelets are also given. Two examples are included to demonstrate the validity and applicability of the approach.


Introduction
A lot of scientific and engineering problems involving fractional phenomenon are already very large and still growing.One of the main advantages of the fractional phenomenon is that the fractional derivatives and fractional integrals provide an excellent approach to the different kinds of physical fields, such as dispersive transports in amorphous semiconductors, tracer transfer in underground water, and seepage in soil or rocks [1][2][3][4].In recent years, more and more researchers are finding that a variety of dynamical problems exhibit fractional order behavior.This indicates that variable order calculus is an effective mathematical framework to describe the complex dynamical problems [5][6][7].Many of the numerical methods using various fractional derivative operators and integral operators for solving fractional differential equations have been proposed.Podlubny [8] used the Laplace transform method to solve the fractional partial differential equations with constant coefficients.Odibat and Momani [9] applied generalized differential transform method to solve the numerical solution of linear partial differential equations of fractional order.Zhang [10] discussed a practical implicit method to solve a class of initial boundary value spacetime fractional convection-diffusion equations with variable coefficients.Zhuang et al. [11] proposed explicit and implicit Euler method for the variable order fractional advectiondiffusion equation.
Bratu's problem is also discussed in all kinds of applications, such as chemical reaction theory, the fuel ignition model of the thermal combustion theory, and nanotechnology [12][13][14][15].Both mathematicians and physicists have devoted a lot of effort to Bratu's problem.In [16], Syam and Hamdan presented the Laplace Adomian decomposition method for solving Bratu's problem.Wazwaz [17] proposed the Adomian decomposition method for solving Bratu's problem.Aksoy and Pakdemirli [18] had solved Bratu-type equation of new perturbation iteration solutions.

Definitions and Properties of Fractional Operator
Here we just recall the most typical definitions which are easy to use in physics.The Caputo fractional differential operator   of order  > 0 is defined as [8] The Riemann-Liouville fractional integration of order  > 0 is defined as where  is a positive integer.It has the following two basic properties for  − 1 <  ≤ :

Moreover, we conclude that
This completes the proof.

Operational Matrix of the Fractional Integration.
In this part, we may simply introduce the operational matrix of fractional integration of CAS wavelets; more detailed introduction can be found in [19].
Apart from the CAS wavelets, we consider another basis set of block pulse functions.The set of these functions, over the interval [0, 1), is defined as with a positive integer value for , we suppose  = 2  (2 + 1) in this paper.
Taking a closer look at Table 1 and Figures 1-3, with  increasing, we find that the approximate solutions converge to the exact solution.It is evident from Figure 4 that, as  close to 1, the numerical solution of the CAS wavelets converges to the exact solution; that is, the solution of fractional integrodifferential equation approaches to the solution of integer order integrodifferential equation.

Conclusion
In this paper, a numerical method is presented by numerical solutions of fractional integrodifferential equations of Bratu type.Taking full advantage of the definition of Caputo type fractional derivative and the properties of CAS wavelet, we  transform the initial problem into a nonlinear algebraic system equation.By solving the nonlinear system, numerical solutions are obtained.The convergence analysis of CAS wavelets and the uniqueness theorem of this equation are proposed.The numerical results show that the approximation is in very good coincidence with the exact solution.
Numerical solution for  = 0.5 Numerical solution for  = 0.75 Numerical solution for  = 1 Exact solution for  = 1

Table 1 :
Absolute errors for different values of , .