An applicaTION of the Exp-funcTION method (EFM) to search for exact soluTIONs of nonlinear partial differential equaTIONs is analyzed. This method is used for the MODIFIED KDV equaTION and the generalized KDV equaTION. The EFM was used to construct peri- odic wave and solitary wave soluTIONs of nonlinear evoluTION equaTIONs (NLEEs). This method is developed for searching exact travelling wave soluTIONs of nonlinear partial differential equaTIONs. It is shown that the Exp-funcTION method, with the help of symbolic computaTION, provides a straightforward and powerful mathematical tool for solving nonlinear evoluTION equaTIONs in mathematical physics and applied mathematics.

The Black-Scholes equaTION is one of the most important mathematical models in opTION pricing theory, but this model is far from market realities and cannot show memory e, ect in the , nancial market. This paper investigates an American opTION based on a time-fracTIONal Black-Scholes equaTION under the constant elasticity of variance (CEV) model, which parameters of interest rate and dividend yield sup-posed as deterministic funcTIONs of time, and the price change of the underlying asset follows a fractal transmission system. This model does not have a closed-form soluTION,hence, we numerically price the American opTION by using a compact di, er-ence scheme. Also, we compare the time-fracTIONal Black-Scholes equaTION under the CEV model with its generalized Black-Scholes model as ,= 1 and ,= 0. Moreover, we demonstrate that the introduced di, erence scheme is uncondiTIONally stable and convergent using Fourier analysis. The numerical examples illustrate the e, ciency and accuracy of the introduced di, erence scheme.

In this article, a simple and new algorithm is proposed, namely the MODIFIED variaTIONal iteraTION algorithm-I (mVIA-I), for obtaining numerical soluTIONs to different types of fifth-order Korteweg de-Vries (KDV) equaTIONs. In order to verify the precision, accuracy and stability of the mVIA-I method, generated numerical results are compared with the Laplace decomposiTION method, Adomian decomposiTION method, Homotopy perturbaTION transform method and the MODIFIED Adomian decomposiTION method. Comparison with the menTIONed methods reveals that the mVIA-I is computaTIONally attractive, excepTIONally productive and achieves better accuracy than the others.

In this paper, by applying the non-classical symmetry method, nonclassical symmetries of the Kodryashov-Sinleschikov (K-S) and MODIFIED Korteweg-de Vries-Zaharov-Kuznetsov (mKDV-ZK) equaTIONs are obtained. Apart from classical symmetries, this theory can be effective in finding a few other soluTIONs for a system of PDEs and ODEs. Also, non-classical symmetries of a system of PDEs can be applied to reduce the number of independent variables. By adding the invariance surface condiTION to the assumed equaTIONs and applying the classical symmetry method for them, non-classical symmetries are calculated. Finally, some of the group invariant soluTIONs and the similarity reduced equaTIONs associated to non-classical symmetry are obtained.

The meshless Fragile Points method (FPM) is applied to find the numerical soluTIONs of the Schrödinger equaTION on arbitrary domains. This method is based on Galerkin’s weak-form formulaTION, and the generalized finite difference method has been used to obtain the test and trial funcTIONs. For partiTIONing the problem domain into subdomains, Voronoi diagram has been applied. These funcTIONs are simple, local, and discontinuous poly-nomials. Because of the discontinuity of test and trial funcTIONs, FPM may be inconsistent. To deal with these inconsistencies, we use numerical flux correcTIONs. Finally, numerical results are presented for some exam-ples of domains with different geometric shapes to demonstrate accuracy, reliability, and efficiency.

We consider a , rst-order partial di, erential equaTION with constant irreversible co-e, cients in a Banach space in the regular case. The equaTION is split into equaTIONs in subspaces, in which non-degenerate subsystems are obtained. We obtain an ana-lytical soluTION of each system with Showalter-type condiTIONs. Finally, an example is given to illustrate the theoretical results.

In this article, we constructed a numerical scheme for singularly perturbed time-delay reacTION-diffusion problems. For the discretizaTION of the time derivative, we used the Crank-Nicolson method and a hybrid scheme, which is a combinaTION of the fourth-order compact difference scheme and the cen-tral difference scheme on a special type of Shishkin mesh in the spatial di-recTION. The proposed scheme is shown to be second-order accurate in time and fourth-order accurate with a logarithmic factor in space. The uniform convergence of the proposed scheme is discussed. Numerical investigaTIONs are carried out to demonstrate the efficacy and uniform convergence of the proposed scheme, and the obtained numerical results reveal the better per-formance of the present scheme, as compared with some existing methods in the literature.