They implemented the new proposed procedure by using a quotient trigonometric function expansion method and also proved that the introduced method could be easily applied to solve other nonlinear differential equations.
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Recently, Wang et al. For instance, Zhang et al. For finding the new exact solutions of NLEEs, it is important to present various method and ansatz, but it seems to be more important how to obtain more new exact solutions to NLEEs under the known method and ansatz. The generalized Riccati Equation 5 has twenty seven solutions Zhu, as follows:.
Step 3: To determine the positive integer m , substitute Equation 4 along with Equation 5 into Equation 3 and then consider homogeneous balance between the highest order derivatives and the nonlinear terms appearing in Equation 3. Step 5: Suppose the value of the constants a n p , q , r and V can be determined by solving the set of algebraic equations obtained in step 4.
Since the general solutions of Equation 5 are known, substituting, a n p , q , r and V into Equation 4 , we obtain new exact traveling wave solutions of the nonlinear evolution Equation 1. Some exact solutions of the DSW equation were found in the literature. In general, the solutions of the DSW equation have been obtained by means of an ansatz method.
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It is to be highlighted that Marinca et. Moreover, a detailed study is made by Yang on local fractional differential equations and its Applications, Local Fractional Functional Analysis and its Applications along with local fractional variation iteration and local fractional Fourier series methods. He has also given a comprehensive analysis of Asymptotic methods for solitary solutions and compactons.
Let us consider the DSW equation:. Now, we use the wave transformation Equation 2 into Equations 6 and 7 , which yield:. Therefore, solution Equations 10 and 11 take the form respectively. By means of Equation 5 , Equations 12 and 13 can be rewritten respectively as,.
Applications of the Expansion Method | Taylor & Francis Group
Setting each coefficient of these polynomials to zero, we obtain a set of simultaneous algebraic equations for a 0 , a 1 , a 2 , b 0 , b 1 , p , q , r and V as follows:. Solving the over-determined set of algebraic equations by using the symbolic computation software, such as, Maple, we obtain. Now on the basis of the solutions of Equation 5 , we obtain some new types of solutions of Equations 6 and 7. Because of the arbitrariness of the parameters b 1 , p , q and r in the above families of solution, the physical quantities u and v might possess physically significant rich structures.
Graph is a powerful tool for communication and describes lucidly the solutions of the problems. Therefore, some graphs of the solutions are given below. It is important to point out that the obtained solutions have not been reported in the previous literature. The new type of traveling wave solutions found in this article might have significant impact on future research.
We assured the correctness of our solutions by putting them back into the original Equations 6 and 7.
Start a New Search:. Record 1 of 1. Simple explicit expressions are obtained for the surface Mach numbers and surface pressures in the special case of slender bodies.
Waves, Wavelets and Fractals
Advanced Search. Article Navigation. Close mobile search navigation Article Navigation. Volume Article Contents. Navier—Stokes equations for the large Reynolds number.
Thermal Expansion of Solids and Liquids
Steady vortex solutions. Linear perturbation and stability of the disk. Wave patterns formed by the perturbations. Summary and remarks. Appendix: Derivation of 1.
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Application of the viscosity-expansion method to a rotating thin fluid disk bound by central gravity Koichi Takahashi. Oxford Academic.