Wet chemistry was used to produce copper oxide nanoparticles (CuO NPs). The results indicated that most nanoparticles were bacillus-shaped and relatively uniform in size (less than 30 nm). The effect of synthesized CuO NPs on wheat (Triticum aestivum L.) germination and growth parameters was studied and compared to bulk Cu. The results showed that no significant difference was obtained in germination rate among all treatments. Bulk Cu additions significantly affect the mean germination rate and mean germination time. On the contrary, germinability was significantly affected by CuO NPs additions. Seed vigor index was calculated to demonstrate the superior treatment in wheat germination parameters, and the results confirmed that 0.1 mg L⁻¹ of CuO NPs could be successfully used to improve wheat seed germination. Moreover, the general average Cu concentrations in the plant tissue were 139 and 103 mg kg⁻¹ dry weight for bulk and CuO NPs, respectively, indicating the dissolution behavior of CuO NPs. The addition of CuO NPs (0.1 mg L⁻¹) promotes chlorophyll formation equal to 0.5 mg L⁻¹ of the bulk Cu addition. This means using nanoparticles as fertilizer could reduce 80% of traditional fertilizers. Nonetheless, Cu additions in both forms (NPs and bulk) reduce root growth substantially compared to control. The effective toxic dose (EC₅₀) for bulk Cu and CuO NPs was 0.37 mg L⁻¹ and 0.94 mg L⁻¹, respectively. The results indicated that approximately 2.5 times CuO NPs concentration is equal to the toxicity dose of bulk Cu due to lowered CuO NPs dissolution. Our study showed that Cu phytotoxicity is a non-nanosized effect and showed that plant-induced changes under environmentally real conditions should be considered when measuring the dissolution of CuO NPs near wheat plant roots. This study implies that using nano-CuO as a micronutrient amendment has a potential benefit rather than the soluble Cu salt for plant growth.

This paper presents a new algorithm for resolving linear and non-linear second-order Robin boundary value problems (BVPS) and the Bratu-type equations in one and two dimensions using spectral approaches. Basis functions according to second-kind shifted and modified shifted Chebyshev polynomials that comply with the Robin conditions are created. It has produced operational matrices for its derivatives. The provided solutions are the result of applying the collocation and tau approaches. These methods convert the problem dictated by its boundary conditions into a system of linear or non-linear algebraic equations that may be solved using any suitable numerical solver. Convergence analysis has been provided and it accords with the numerical results. Six numerical problems are provided to investigate and demonstrate the practical utility of the suggested method. The current results show that our method outperforms the previous methods in terms of accuracy which are presented in tables and figures.

In this article, we evaluated the approximate solutions of one-dimensional variable-order space-fractional diffusion equations (sFDEs) by using a collocation method. This method depends on operational matrices for fractional derivatives and the integration of generalized Fibonacci polynomials. In this method, a Caputo fractional derivative of variable order is applied. Some properties of these polynomials (using boundary conditions) are presented to simplify and transform sFDEs into a system of equations with the expansion coefficients of the solution. Also, we discuss the convergence and error analysis of the generalized Fibonacci expansion. Finally, we compare the obtained results with those obtained via the other methods.

In the current manuscript, we implement the collocation method to obtain an approximate solution of one-dimensional time-fractional convection equation. The operational matrices of Pell polynomials are applied to solve the fractional partial differential equations. In the Caputo sense, we describe the time-fractional derivatives. So, this algorithm allows us to transform the fractional differential equation with initial (boundary) conditions into a system of algebraic equations in a good form. The convergence and error analysis of Pell polynomials are carefully investigated, and some examples are explored to show the comparison and efficiency of our method with others.

Herein, we use the generalized Lucas polynomials to find an approximate numerical solution for fractional initial value problems (FIVPs). The method depends on the operational matrices for fractional differentiation and integration of generalized Lucas polynomials in the Caputo sense. We obtain these solutions using tau and collocation methods. We apply these methods by transforming the FIVP into systems of algebraic equations. The convergence and error analyses are discussed in detail. The applicability and efficiency of the method are tested and verified through numerical examples.