Boundary layer flow on bioconvection model of nanoparticles by non-singular and non-local Kernal with neural network

The free convective flow of heat transport with nanofluids over a plate must be studied in order to enhance thermal technologies and increase the efficiency of heat transfer in engineering system. The present article introduces the study of nanoparticles with the base fluid of water. Heat transfer on fluid with the multiple effects are taken into account. The new work uses the combined action of mass transport and heat over a plate by using Fourier’s and Fick’s laws to achieve novel outcomes. The governing fractional partial differential equations are solved using Levenberg-Marquardt backpropagation algorithm-trained artificial neural networks. Training data are generated analytically through the Laplace transform, with a 15% for testing, 70% for data and remaining 15% for validation. The proposed model demonstrates high predictive accuracy, achieving a mean squared error below <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mn>10</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> . Sensitivity analysis reveals that the fractional parameter reduces velocity fluid, while concentration and temperature respond strongly to chemical reactions and thermal conditions. The main contribution of this work lies in combining fractional calculus with ANN-based optimization to provide a reliable, efficient tool for modeling and enhancing heat transfer in nanofluid-based engineering systems. The Nu and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>f</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> for ordinary fluid ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>ϕ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> ) and Nanofluid ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>ϕ</mml:mi> <mml:mo>≠</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> ) by using the values of Pr from 6.2 to 1000.