A computational collocation approach for solving Fredholm integral equations in oil and gas exploration

Fredholm integral equations of the second kind are fundamental for modelling non-local interactions in subsurface fluid flow, reservoir pressure communication, and seismic wave propagation. While analytical solutions are limited to simplified cases, practical petroleum engineering problems require robust numerical methods. This study develops a polynomial collocation scheme for solving representative Fredholm integral equations associated with oil and gas exploration models in Nigeria. The unknown solution is approximated by monomial basis functions 𝜑_𝑗(𝑥)=𝑥^𝑗−1 on[0,1], and the governing equation is enforced at collocation points to yield a linear algebraic system. MATLAB R2024a implementations for polynomial kernel K(x,t)=x+t and oscillatory kernel K(x,t)=sin(x-t) achieve maximum absolute errors of 3.20 × 10⁻⁵ with n=6. Error analysis confirms O(n⁻⁴) convergence, while comparison with Galerkin methods shows a 57% reduction in assembly time due to explicit kernel integration. The results establish monomial collocation as an accurate, stable, and field-implementable framework for reservoir simulation and seismic interpretation.