Hybrid Upadhyaya Transform and Power Series Technique for Addressing Nonlinear Volterra Equations of the First Kind
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Abstract
In Mathematics, biology, physics and engineering, nonlinear Volterra integral equations (NVIEs) of the first kind are frequently encountered when modelling dynamic systems. However, because of their ill-posed nature and nonlinear terms, they present considerable difficulties. This work presents a hybrid methodology that combines a power series expansion with the Upadhyaya transform, a flexible tool from the Laplace family, building on recent developments in integral transforms. This combination resolves nonlinearities through systematic coefficient matching in the series domain and simplifies the handling of convolution kernels via the transform. We describe the fundamentals of the approach, show how it can be applied to four benchmark problems taken from earlier research, and expand it to a new case involving trigonometric nonlinearity. With an emphasis on computational clarity and verification, each example is broken down step-by-step. The results show that the hybrid approach outperforms standalone methods in terms of flexibility and ease, producing exact solutions when feasible and convergent approximations otherwise. There is potential for this method to be applied more widely in solving integral models in the real world.
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References
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