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  • [ X ]
    Öğe
    An efficient method for the analytical study of linear and nonlinear time-fractional partial differential equations with variable coefficients
    (Samara State Technical Univ, 2023) Liaqat, M. I.; Akgul, A.; Prosviryakov, E. Yu.
    The residual power series method is effective for obtaining approximate analytical solutions to fractional-order differential equations. This method, however, requires the derivative to compute the coefficients of terms in a series solution. Other well-known methods, such as the homotopy perturbation, the Adomian decomposition, and the variational iteration methods, need integration. We are all aware of how difficult it is to calculate the fractional derivative and integration of a function. As a result, the use of the methods mentioned above is somewhat constrained. In this research work, approximate and exact analytical solutions to time-fractional partial differential equations with variable coefficients are obtained using the Laplace residual power series method in the sense of the Gerasimov-Caputo fractional derivative. This method helped us overcome the limitations of the various methods. The Laplace residual power series method performs exceptionally well in computing the coefficients of terms in a series solution by applying the straightforward limit principle at infinity, and it is also more effective than various series solution methods due to the avoidance of Adomian and He polynomials to solve nonlinear problems. The relative, recurrence, and absolute errors of the three problems are investigated in order to evaluate the validity of our method. The results show that the proposed method can be a suitable alternative to the various series solution methods when solving time-fractional partial differential equations.
  • [ X ]
    Öğe
    Approximate analytical solutions of the nonlinearfractional order financial model by two efficientmethods with a comparison study
    (Samara State Technical Univ, 2024) Liaqat, M. I.; Khan, A.; Irshad, A.; Akguel, A.; Prosviryakov, E.
    The financial system has become prominent and important in globaleconomics, because the key to stabilizing the economy is to secure or controlthe financial system or market. The goal of this study is to determine whether or not the approximateanalytical series solutions obtained by the residual power series method andElzaki transform decomposition method of the fractional nonlinear financialmodel satisfy economic theory. The fractional derivative is used in the senseof the Caputo derivative. The results are depicted numerically and in figures that show the behav-ior of the approximate solutions of the interest rate, investment demand,and price index. Both methods yielded results in accordance with economictheory, which established that researchers could apply these two methods tosolve various types of fractional nonlinear problems that arise in financialsystems