This study addresses the solvability and explicit solutions of various boundary value problems (BVPs) for inhomogeneous equations within the upper half unit disc. Specifically, we investigate the Dirichlet, Neumann, and mixed Dirichlet-Neumann problems for the inhomogeneous Cauchy-Riemann and Bitsadze equations. By employing analytical techniques and function space theory, we establish necessary and sufficient conditions for the existence of solutions. Furthermore, explicit solution formulas are derived under these solvability criteria, providing a constructive approach to solving such BVPs. The significance of this research lies in its contribution to the broader theory of BVPs in complex domains. The results obtained not only extend classical boundary conditions but also offer a systematic framework for dealing with higher-order equations. The interplay between different boundary conditions is explored in detail, revealing new insights into the structure of solutions and their dependence on boundary data. Beyond the theoretical implications, our findings have potential applications in mathematical physics, fluid dynamics, and engineering, where such problems frequently arise in modeling physical phenomena. Future research may further extend these results to more general domains and nonlinear equations, enriching the field of complex analysis and partial differential equations.
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