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This paper presents the Mellin transform for the solution of the fractional order equations. The Mellin transform approach occurs in many areas of applied mathematics and technology. The Mellin transform of fractional calculus of different flavours; namely the Riemann-Liouville fractional derivative, Riemann-Liouville fractional integral, Caputo fractional derivative and the Miller-Ross sequential fractional derivative were obtained. Three illustrative examples were considered to discuss the applications of the Mellin transform and its fundamental properties. The results show that the Mellin transform is a good analytical method for the solution of fractional order equations.

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This paper presents the Mellin transform method for the valuation of some vanilla power options with non-dividend yield. This method is a powerful tool used in the valuation of options. We extend the Mellin transform method proposed by Panini R. and Srivastav R.P. [15] to derive the price of European and American power put options with non-dividend yield. We also derive the fundamental valuation formula known as the Black-Scholes model using the convolution property of the Mellin transform method. To provide a sufficient numerical analysis, we compare the results generated by the Mellin transform method for the valuation of American power put option for n = 1 which pays no dividend yield to two other numerical methods namely Crank Nicolson finite difference method [2] and binomial model [3] for options valuation against Black-Scholes analytical pricing formula [1]. The numerical experiment shows that the Mellin transform method is efficient, easy to implement, agree with the values of Black-Scholes [1], Crank Nicolson finite difference method [2] and binomial model [3]. Hence the Mellin transform method is a better alternative method compared to the Crank Nicolsion finite difference and binomial model for the valuation of some vanilla power options.

This paper presents 2-step p-th order (p = 2,3,4) multi-step methods that are based on the combination of both polynomial and exponential functions for the solution of Delay Differential Equations (DDEs). Furthermore, the delay argument is approximated using the Lagrange interpolation. The local truncation errors and stability polynomials for each order are derived. The Local Grid Search Algorithm (LGSA) is used to determine the stability regions of the method. Moreover, applicability and suitability of the method have been demonstrated by some numerical examples of DDEs with constant delay, time dependent and state dependent delays. The numerical results are compared with the theoretical solution as well as the existing Rational Multi-step Method2 (RMM2).