Implementing the RDTM to the

Implementing the RDTM to the two dimensional telegraph Eq. (1), we have the following expressionNow implementing the aforesaid method to the initial conditions (2), we getFrom above two equations we get the values of etc. Applying the differential inverse reduced transform of , one can obtain the approximate solution for and given by

RDTM for three dimensional telegraph equation
Implementing the aforesaid method to the three dimensional telegraph Eq. (4), we get the following equationNow applying the method to the initial conditions (5), we haveFrom above two equations we get the values of etc. Using the differential inverse reduced transform of , we have the approximate solution for and as follows

Computational illustrations
In this dihydrochloride section, the method explained in Section 2 is described by taking several examples of both linear and nonlinear 2D and 3D telegraph equations to validate the efficiency and reliability of the aforesaid technique.Implementing the RDTM to Eq. (11), we get the following relationUsing the aforesaid method to the initial conditions (12), we haveUsing Eq. (14) in Eq. (13), we have the following values successively asUsing the differential inverse reduced transform of , we obtain the expressionThe solution (16), in closed form, is expressed as followsApplying the aforesaid technique to Eq. (18), we obtain the following recurrence formulaUsing the described method to the initial conditions (19), we getUsing Eq. (21) in Eq. (20), one can get the following values successively asUsing the differential inverse reduced transform of , we haveThe solution (23), in closed form, can be given byApplying the aforesaid technique to Eq. (25), we obtain the following iterative expression:Applying the RDTM to the initial conditions (40), we obtainUsing Eq. (28) in Eq. (27), we get the following values successively asUsing the differential inverse reduced transform of , one can getThe solution (30), in closed form, is given byImplementing the aforesaid technique to Eq. (32), we obtain the following iterative expression:Using the aforesaid scheme to the initial conditions (33), we haveUsing Eq. (35) in Eq. (34), we obtain values successively asUsing the differential inverse reduced transform of , we haveThe solution (37), in closed form, is given as follows

Conclusions

Introduction
Zn–Al based composites have continued to find relevance in several technological applications (Can Kurnaz, 2003). The Zn–Al alloys, which serve as the matrix for this class of MMCs, are known for their good combination of physical, mechanical and technological properties. High strength, excellent castability, good machinability, low melting point and good tribological properties, as well as low manufacturing cost are among its notable characteristics (Zhu et al., 2003; Savaskan and Hekimoglu, 2014). They have shown satisfactory service performance when used for the design of components such as bearings, dies, punches and seals which require high mechanical and wear resistance (Bobic et al., 2009). The inability of ZA based alloys to work effectively above operating temperatures of 80°C has been a sour limitation to its application for several other purposes. In order to take advantage of its base properties, reinforcing ZA alloys with ceramic materials has been explored (Xu et al., 2006). The use of reinforcements such as silicon carbide (SiC) and alumina (Al2O3) has resulted in marked improvement in hardness, strength, specific strength, wear and creep resistance of Zn–Al based composites (Mitrović et al., 2007; Bobic et al., 2014). The problem of machinability of Zn–Al based composites has been improved on (without any deleterious effect on mechanical and tribological properties) by the use of graphite as complementing reinforcement to SiC and Al2O3 (Mitrović et al., 2011). The development of low cost Zn based composites with the use of conventional reinforcements such as SiC and alumina and agro waste ashes as complements has been marginally reported on in the literature (Alaneme et al., 2014). The use of such a hybrid combination of reinforcing materials has been well explored for Al based metal matrix composites (MMCs) with great technical promise reported (Alaneme et al., 2013a,b; Escalera-Lozano et al., 2008). For Zn–Al based MMCs this is a virgin area with the potential of offering reduced composite production cost, additional channel for agrowaste recycling, while still maintaining the technical efficiency and performance levels of conventional Zn–Al based composites. In the present study, the mechanical properties of Zn–27Al based composites reinforced with SiC, rice husk ash (RHA), and graphite (Cg) are reported in this paper.