Solving the Prandtl ’ s equation by the modified Adomian decomposition method

This paper aims at modifying the Adomian decomposition method. Thus, it is expected that a class of hypersingular integral equations of the second kind namely Prandtls equation, be solved. To illustrate the procedure, different examples and problems are allotted to confirm the effectiveness, accuracy and validity of the modified method. All results considered, one can say that MADM promises the exact solution .It also can be conveniently used. That is, MADM is a simple but precise procedure that does not perform the complicated function in collocation methods based on polynomial.


Introduction
In the field of applied Mathematics, of course, due to the nature of hypersingular integral equation, it is selected in such a way that it becomes as an effective measure.In structuring an application to solve a large class of mixed boundary value problems is being promised in mathematical physics.It is imperative to mention that the crack problems in fracture mechanics or water wave scattering problems including obstacles, diffraction of electromagnetic waves and aerodynamics problems [2,6,7,10] would be decreased to hypersingular integral equations in single or disjoint multiple intervals.A simple approximation method is being practiced to solve a general hypersingular integral equation of the first kind where the kernel includes a hypersingular part and a regular part [8].
The approximate solution of a class of singular integral equations of the second kind is best achieved using a method based on polynomial approximation [9].A hypersingular integral equation in two intervals has been investigated by Dutta and Banerjea [4] through conducting the solution of Cauchy type singular integral equations in two disjoint intervals.Gori et al. [5] has investigated a quadrature rule based on using suitable refinable quasi-interpolatory operators, for the numerical evaluation of Hadamard finite-part integrals.Chen and Zhou [3] constructed an effectual method to achieve a solution to hypersingular integral equation of the first kind in a reproducing kernel space in order to make it possible to eliminate the singularity of the equation Here we proceed with the hypersingular integral equations of the second kind on the finite interval (1, 1) subject to v(±1) = 0. Eq. (1.1) represents a generalized state for oval wing of Prandtl's equation, where α (> 0) is a known value, u(r) and v(r) are known and unknown functions, respectively.Eq. (1.1) indicates to Hadamard finite part [10].
The exact solution of Eq. (1.1) can be accomplished as follows 3) The exact solution to Eq. (1.3) can not be conveniently approximated [6,7].As a matter of fact, using a simple approximating polynomial for g(t) in [9] and then turning it to a differential problem of Riemann Hilbert on the interval (−1, 1) in [1] the exact solution to Eq. (1.1) will be promised.In this paper, we are more concerned to establish an analytical solution to Eq. (1.1) by employing MADM.To achieve this goal, the next sections may prove helpful.
Taking into account MADM we select u 1 (r) = 4π π+2 and u 2 (r) = 2π 2 π+2 .Now, from Eq. (2.12) there could be Finally, solution to the integral equation Eq. (3.13) is accomplished as following which is chosen as the exact solution of the integral equation.
Example 3.2.it is recommended to consider the following hypersingular integral equation of the second kind [10] where the exact solution is Then, it is written

.31)
Here u(r) = 5r 3 − 7r which is a polynomial of degree three, then we select

Conclusion
In this paper, a class of hypersingular integral equations of the second kind (Prandtls equation) was placed to be solved.Thus, we attempted at introducing a modified Adomian decomposition method.Besides, there were some solutions to the included examples.Since the numerical results appeared desirable, there is now a fairly good agreement among researchers that the features of MADM such as effectiveness and validity direct their attention to apply it to solve the hypersingular integral equations.