Chein-Shan Liu, Chung-Lun Kuo^*

Digital Engineering and Digital Twin, Vol.2, pp. 145-167, 2024, DOI:10.32604/dedt.2024.042804 - 31 December 2024

Abstract A new concept of projective solution is introduced for the second-order linear partial differential equations (PDEs) endowed with constant coefficients. In terms of a projective variable the PDE is transformed to a second-order ordinary differential equation (ODE) with constant coefficients at the first time. The characteristic form appears as the coefficient preceding the second-order derivative term. Depending on the characteristic form and coefficients we can derive various parameters-dependent particular solutions, which can be adopted as the bases to expand the solution. The Helmholtz and wave equations are solved by the projection method. We project the… More >

Bilal Khan^1,*, Zhi-Guo Liu¹, Nazar Khan², Aftab Hussain³, Nasir Khan⁴, Muhammad Tahir²

CMES-Computer Modeling in Engineering & Sciences, Vol.131, No.3, pp. 1465-1477, 2022, DOI:10.32604/cmes.2022.016940 - 19 April 2022

Abstract By using the basic (or q)-Calculus many subclasses of analytic and univalent functions have been generalized and studied from different viewpoints and perspectives. In this paper, we aim to define certain new subclasses of an analytic function. We then give necessary and sufficient conditions for each of the defined function classes. We also study necessary and sufficient conditions for a function whose coefficients are probabilities of q-Poisson distribution. To validate our results, some known consequences are also given in the form of Remarks and Corollaries. More >

Rabha W. Ibrahim^1,*, Ahmed M. Ajaj², Nadia M.G. Al-Saidi³, Dumitru Balean^4,5,6

CMES-Computer Modeling in Engineering & Sciences, Vol.130, No.1, pp. 221-232, 2022, DOI:10.32604/cmes.2022.018348 - 29 November 2021

Abstract The science of strategy (game theory) is known as the optimal decision-making of autonomous and challenging players in a strategic background. There are different strategies to complete the optimal decision. One of these strategies is the similarity technique. Similarity technique is a generalization of the symmetric strategy, which depends only on the other approaches employed, which can be formulated by altering diversities. One of these methods is the fractal theory. In this investigation, we present a new method studying the similarity analytic solution (SAS) of a 3D-fractal nanofluid system (FNFS). The dynamic evolution is completely… More >