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This paper studies the numerical algorithm of stochastic control problems in investment optimization. Investors choose the optimal investment to maximize the expected return under uncertainty. The optimality condition, the Hamilton–Jacobi–Bellman (HJB) equation, satisfied by the value function and obtained by the dynamic programming method, is a partial differential equation coupled with optimization. One of the major computational difficulties is the irregular boundary conditions presented in the HJB equation. In this paper, two mesh-free algorithms are proposed to solve two different cases of HJB equations with regular and irregular boundary conditions. The model of optimal investment under uncertainty developed by Abel is used to study the efficacy of the proposed algorithms. Extensive numerical studies are conducted to test the impact of the key parameters on the numerical efficacy. By comparing the numerical solution with the exact solution, the proposed numerical algorithms are validated. [ABSTRACT FROM AUTHOR]

Lithium–sulfur (Li–S) batteries have received considerable attention due to their high theoretical specific capacity (1675 mAh g–1) and environmental friendliness. Whereas, the development of Li–S batteries has been limited by the shuttle effect and volume expansion. Binders play a key role to alleviate the above issues as an essential part of the Li–S batteries. However, there are few studies about the application of vegetable oils for preparing Li–S batteries binders. Herein, a cross–linked reactive castor oil–based binder (CAB) initiated by UV light is reported. The robust covalent cross–linking network relieves the volume expansion and ensures the stability of the electrode while numerous polar groups such as –C O and –NH– suppress the shuttle effect by immobilizing polysulfides. Therefore, the Li–S batteries with CAB binder exhibits excellent cycling stability (an initial discharge specific capacity of 1037 mAh g–1, 510 mAh g–1 after 500 cycles), and a remarkable areal capacity of 3.8 mAh cm–2 after 70 cycles even though a high sulfur loading (5.3 mg cm–2). The reactive eco–vegetable oil–based binder confirms the feasibility of UV–curable CAB as the binder for Li–S batteries and contributes to the development of efficient vegetable oil–based, reactive binders with the improved mechanical properties. [Display omitted] [ABSTRACT FROM AUTHOR]