We examine the diffusion equation on the sphere. In this sense, we answer the question of the symmetry classification. We provide the algebra of symmetry and build the optimal system of Lie subalgebras. We prove for one-dimensional optimal systems of Eq. (1. 4), space is expanding Ricci solitons. Reductions of similarities related to subalgebras are classified, and some Exact invariant solutions of the diffusion equation on the sphere are presented.

The Lie symmetry method for differential equations is applied to study the Exact solutions of the acoustic PDE. This study is based on two methods: Kudryashov method and the direct method. Some Exact solutions are found by using the similarity variables extracted from symmetries.

In this paper, European options with transaction cost under some Black-Scholes markets are priced. In fact, stochastic analysis and Lie group analysis are applied to find Exact solutions for European options pricing under considered markets. In the sequel, using the finite difference method, numerical solutions are presented as well. Finally, European options pricing are presented in four maturity times under some Black-Scholes models equipped with the gold asset as underlying asset. For this, the daily gold world price has been followed from Jan 1, 2016 to Jan 1, 2019 and the results of the profit and loss of options under the considered models indicate that call options prices prevent arbitrage opportunity but put options create it.

Using a traveling wave reduction technique, we have shown that Maccari equation, (2+1) -dimensional nonlinear Schrodinger equation, medium equal width equation, (3+1) -dimensional modified KdV–Zakharov–Kuznetsev equation, (2+1) -dimensional long wave-short wave resonance interaction equation, perturbed nonlinear Schro¨dinger equation can be reduced to the same family of auxiliary elliptic-like equations. Then using extended F-expansion and projective Riccati equation methods, many types of Exact traveling wave solutions are obtained.With the aid of solutions of the elliptic-like equation, more explicit traveling wave solutions expressed by the hyperbolic functions, trigonometric functions and rational functions are found out. It is shown that these methods provide a powerful mathematical tool for solving nonlinear evolution equations in mathematical physics. A variety of structures of the Exact solutions of the elliptic-like equation are illustrated.

In the current study, we consider the generalized Pochhammer-Chree equation with a term of order n. Based on the (1/G0)-expansion method and with the aid of symbolic computation, we construct some distinct Exact solutions for this nonlinear model. Various Exact solutions are produced to the studied equation including singular solutions and periodic wave solutions. In addition, 2D, 3D, and contour plots are graphed for all obtaining solutions via choosing the suitable values for the involved parameters. All gained solutions verify the governing equation.

By means of He's homotopy perturbation method (HPM) an approximate solution of velocity field is derived for the flow in straight pipes of non-Newtonian fluid obeying the Sisko model. The nonlinear equations governing the flow in pipe are for-mulated and analyzed, using homotopy perturbation method due to He. Furthermore, the obtained solutions for velocity field is graphically sketched and compared with Newtonian fluid to show the accuracy of this work. Volume flux, average velocity and pres-sure gradient are also calculated. Results reveal that the proposed method is very effective and simple for solving nonlinear equations like non-Newtonian fluids.

In this work a considerable number of papers regarding the cracked beam like structures in the literature are cited. A new and general governing differential equation for eigenvalue analysis of cracked members is derived. Buckling analysis of cracked columns, lateral free vibration of cracked beams, axial free vibration of cracked bars, torsion free vibration of cracked shafts etc. may be considered as special cases. The proposed standard ordinary differential equation is solved and the Exact analytical solutions for eigenvalues and mode shapes of these members are determined. Through customizing the general solutions for special conditions the predefined solutions are obtained and the accuracy and robustness of the present study is verified.