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In this paper, a mathematical model is given that depicts the interactions between cancer cells and viruses in the setting of oncolytic virotherapy. The model is separated into three classes, namely, concentrations of uninfected tumor cells in the population " U$$ \mathcal{U} $$", free virus " V$$ \mathcal{V} $$", and cancerous cells infected " I$$ \mathcal{I} $$". Applying Caputo fractional derivative, the model is fractionalized, and using generalized Bessel polynomials, an optimal problem is solved utilizing Lagrange multipliers method. The results show that the presented method has high accuracy and is suitable for solving the nonlinear systems based on partial differential equations especially tumors models. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a general class of coupled nonlinear systems of variable-order fractional partial differential equations (GCNSV-FPDEs) with initial and boundary conditions. We propose a hybrid method based on new generalized Bernoulli–Laguerre polynomials (GB-LPs) for solving GCNSV-FPDEs. The concept of variable-order fractional derivatives (V-FDs) is employed in the Caputo type. We extract the operational matrices (OMs) of classical and V-FDs of GB-LPs. By utilizing GB-LPs, OMs, and the Lagrange multipliers method, we transform the given GCNSV-FPDE into a system of algebraic equations to be solved. The proposed method yields satisfactory results even with a small number of GB-LPs. We provide a full verification of the method's convergence, and two examples are included to demonstrate its validity and applicability. [ABSTRACT FROM AUTHOR]

Background: Cancer, a complex and deadly health concern today, is characterized by forming potentially malignant tumors or cancer cells. The dynamic interaction between these cells and their environment is crucial to the disease. Mathematical models can enhance our understanding of these interactions, helping us predict disease progression and treatment strategies. Methods: In this study, we develop a fractional tumor-immune interaction model specifically for lung cancer (FTIIM-LC). We present some definitions and significant results related to the Caputo operator. We employ the generalized Laguerre polynomials (GLPs) method to find the optimal solution for the FTIIM-LC model. We then conduct a numerical simulation and compare the results of our method with other techniques and real-world data. Results: We propose a FTIIM-LC model in this paper. The approximate solution for the proposed model is derived using a series of expansions in a new set of polynomials, the GLPs. To streamline the process, we integrate Lagrange multipliers, GLPs, and operational matrices of fractional and ordinary derivatives. We conduct a numerical simulation to study the effects of varying fractional orders and achieve the expected theoretical results. Conclusion: The findings of this study demonstrate that the optimization methods used can effectively predict and analyze complex phenomena. This innovative approach can also be applied to other nonlinear differential equations, such as the fractional Klein–Gordon equation, fractional diffusion-wave equation, breast cancer model, and fractional optimal control problems. [ABSTRACT FROM AUTHOR]

This article introduces a new class of basis functions, namely, the generalized Bernoulli polynomials (GBP). The GBP are adopted for solving nonlinear fractional optimal control problems (NFOCP) generated by nonlinear fractional dynamical systems (NFDS) and boundary conditions (BC). The corresponding operational matrices (OM) of fractional derivatives (FD) expand the solution of the problem in terms of the GBP. The method transforms the NFOCP into systems of nonlinear algebraic equations. First, the state and control variables are approximated by the GBP with unknown coefficients and parameters and substituted in the objective function, NFDS and BC. Then, the Gaussian quadrature rule and the OM of FD allow the formulation of a constrained problem, which is solved using Lagrange multipliers. The accuracy of the method is tested by means of several examples and the results confirm its good performance. [ABSTRACT FROM AUTHOR]

In this paper, we apply transcendental Bernstein series (TBS) for solving reaction–diffusion equations with nonlocal boundary conditions which is the novel approximation tool. To carry out the method, we firstly expand the solution of the system in the term of TBS through the operational matrix scheme. To determine the unknown free coefficients and control parameters appeared in TBS expansion, we define an optimization problem which combines the reaction–diffusion equation with its nonlocal boundary conditions. Then we use the Lagrange multipliers technique for converting the problem under study into a system of algebraic equations. High accuracy and simplicity in reducing the integral boundary conditions are some privileges of the proposed scheme. We emphasize that Bernstein polynomials is the particular case of transcendental Bernstein series. Theoretical discussion about convergence confirms the reliability of the proposed method. Some test problems are chosen to investigate the applicability and computational efficiency. The experimental results confirm that the obtained results are in good agreement with the exact solutions with high rate of convergence. [ABSTRACT FROM AUTHOR]

This paper proposes an optimization method for solving fractional variational problems depending on indefinite integrals, where the fractional derivative is described in the Caputo sense. The method is based on the new basis functions consisting of the transcendental Bernstein series (TBS) and their operational matrices. In the first step, we derive an approximate solution for the problem using TBS with the free coefficients and control parameters. In the second step, we use the fractional operational matrix, with the help of the Lagrange multipliers technique, for converting the fractional variational problem into an easier one, described by a system of nonlinear algebraic equations. The convergence analysis of the method, will be guaranteed by proving a new theorem concerning TBS. Finally, for illustrating the efficiency and accuracy of the proposed technique, several numerical examples are analyzed and the results compared with the analytical solutions or the approximation obtained by other techniques. [ABSTRACT FROM AUTHOR]

This paper deals with an efficient numerical method for solving two-dimensional variable-order fractional optimal control problem. The dynamic constraint of two-dimensional variable-order fractional optimal control problem is given by the classical partial differential equations such as convection-diffusion, diffusion-wave and Burgers' equations. The presented numerical approach is essentially based on a new class of basis functions with control parameters, called generalized polynomials, and the Lagrange multipliers method. First, generalized polynomials are introduced and an explicit formulation for their variable-order fractional operational matrix is obtained. Then, the state and control functions are expanded in terms of generalized polynomials with unknown coefficients and control parameters. By using the residual function and its 2-norm, the under consideration problem is transformed into an optimization one. Finally, the necessary conditions of optimality results in a system of algebraic equations with unknown coefficients and control parameters can be simply solved. Some illustrative examples are given to demonstrate accuracy and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

This paper deals with finding an approximate solution of variable order fractional optimal control problems (V-FOCPs) based on Lagrange multiplier optimization technique in combination with the new operational matrix of variable order (VO) fractional derivatives. To carry out the proposed approach, we firstly develop the well-known Bernstein polynomials to the new series of functions namely transcendental Bernstein series (TBS). Then we implement these basis functions to approximate the solutions of V-FOCPs. In fact, the series expansion in TBS with unknown free coefficients and control parameters is the novel idea for solving the fractional systems with less terms of approximation. The convergence analysis of our proposed method, will be guaranteed by proving a new theorem for the TBS. To investigate the practical computational efficiency and accuracy of the presented method, some numerical examples are provided. The experimental results confirm the applicability of the method and a good agreement between the approximate and exact solutions. [ABSTRACT FROM AUTHOR]