Equivalent transformation of differential and integral equations in hyperbolic plane

Hyperbolic numbers, serving as an extension of real numbers, exhibit an algebraic structure akin to complex numbers, constituting commutative rings with zero divisors that stem from two real numbers. This paper delves into the exploration of the equivalence relation between the solutions of differential equations and integral equations within the realm of the hyperbolic plane. The pivotal finding of this study reveals that the solutions of hyperbolic differential equations and integral equations are inherently equivalent. Such a discovery not only solidifies the groundwork for delving into hyperbolic differential equations but also injects a renewed impetus into the advancement of physics. By bridging the gap between differential and integral equations in the hyperbolic domain, our research paves the way for a deeper understanding of hyperbolic numbers’ mathematical structures and their applications in the physical sciences, thereby opening up new avenues for exploration and discovery.