In this paper, generalized aspects of least square homotopy perturbations are explored to treat the system of non-linear fractional partial differential equations and the method is called as generalized least square homotopy perturbations (GLSHP).The concept of partial fractional Wronskian is introduced to detect the linear independence of functions depending on more than one variable through Caputo fractional calculus. General theorem related to Wronskian is also proved. It is found that solutions converge more rapidly through GLSHP in comparison to classical fractional homotopy perturbations. Results of this generalization are validated by taking examples from nonlinear fractional wave equations.Mathematics Subject Classification. 35Qxx, 65Mxx, 65Zxx.In above equations, φ (x,t), ψ(x,t), f i (t) and g i (t) for i = 1, 2 are given functions. Wei [16] analyzed a local finite difference scheme for the diffusion-wave equation of fractional order. A fourth order compact scheme which preserve energy was devised by Diaz et al. [17] for the solution of fractional nonlinear wave equations. Weak solutions for time fractional diffusion equations were examined by Yamamoto [18]. Other schemes which prescribe the information regarding the solution of fractional wave equations can be studied from [19]-[27].In previous years, scientists and applied mathematicians have been attracted towards modifications of classical homotopy perturbations in order to achieve accelerated accuracy. Some modifications can be seen in [28] and [29]. A variational form for homotopy perturbation iteration method was developed for fractional diffusion equation by Guo et al. [30]. Recently, Constantin and Caruntu [31] presented least square homotopy perturbations for non-linear ordinary differential equations.But this technique was not suitable to handle fractional partial differential equations as convergent solutions are not expected.Therefore, our main objective is to generalize this idea of the coupling of fractional homotopy perturbations and least square approximations, and to propose fractional partial Wronskian.The paper is organized in the following sequence: In section 2, basic definitions of fractional calculus are presented.The definition and theory of fractional partial Wronskian is also developed. In section 3, basic theory of generalized least square homotopy perturbations are proposed together with necessary definitions and lemma. Section 4 deals with numerical examples. Lastly conclusion is derived. Fundamentals of fractional calculusWe provide some basic definitions, properties and theorem related to fractional calculus which will be used at later stage C(0, ∞), and is said to be in the space C n µ if and only if f n ∈ C µ , n ∈ N .Definition 2.2. The (left sided) Riemann-Liouville fractional integral operator of order α 0 for a function f ∈ C µ , µ −1 is defined as:α > 0, t > 0, β 2 are linear operators, N , M are non-linear operators and u(x,t), v(x,t) are unknown functions. Whereas B 1 , B 2 are boundary operators, f (x,t) and g...

In this problem, simultaneous effects of Joule and viscous dissipation in three-dimensional flow of nanoliquid have been addressed in slip flow regime under time dependent rotational oscillations. Silver nanoparticles are submerged in the base fluid (water) due to their chemical and biological features. To increment the novelty, effects of cubic autocatalysis chemical reactions and radiative heat transfer have been incorporated in the related boundary layer equations. Dimensionless partial differential system is solved by employing the proposed implicit finite difference approach. Convergence conditions and stability criteria are obtained to ensure the convergence and accuracy of solutions. A comparative analysis is proposed for no-slip nanofluid flow (NSNF) and slip nanofluid flow (SNF). Variations in skin-friction coefficients, Sherwood and Nusselt numbers against physical parameters are tabulated. It is investigated that velocity slip and temperature jump significantly control drag forces and rate of heat transfer.

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In this paper, generalized aspects of least square homotopy perturbation are presented to treat the system of nonlinear fractional partial differential equations (FPDEs), and the method is called as generalization of least square homotopy perturbation (GLSHP). The concept of fractional partial Wronskian is introduced to detect the linear independence of functions depending on more than one variable through Caputo fractional calculus. Definition of fractional partial Wronskian is provided along with linear independence theorem. It is found that solutions converge more rapidly through GLSHP in comparison to classical fractional homotopy perturbation method (FHPM). Outcomes of this generalization are validated by taking examples from nonlinear fractional wave equations.

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