Distributional Solutions to Boundary Value Problems Using Fourier Series

This article explores the application of Fourier collection strategies in fixing complicated boundary charge issues (BVPs) involving fractional derivatives, distributional coefficients, and vector-valued distributions. By leveraging cutting-edge improvements in fractional calculus, tempered distributions, and multidimensional Fourier techniques, a whole framework is developed to address the stressful situations posed by using irregularities, singularities, and non-neighborhood operators in BVPs. The proposed method transforms differential operators into algebraic expressions inside the frequency place, allowing inexperienced and correct answers for issues which may be computationally high priced for conventional numerical strategies. Key consequences display the efficacy of Fourier collection in handling fractional operators, taking pictures the outcomes of distributional coefficients, and solving actual-international problems such as fractional warmness conduction and wave propagation with singular assets. While the technique exhibits speedy convergence for clean forcing phrases, challenges which includes Gibbs phenomena for non-smooth inputs and computational complexity in multidimensional domain names are mentioned. This have a test highlights the flexibility and computational overall performance of Fourier collection strategies, offering a basis for destiny studies in mathematical physics, engineering, and accomplished mathematics