## 1 Introduction  
The computational interpretation of convection in elliptic conduits is an important and worthy topic of recent interest for many researchers. A huge mathematical literature exists on the distinct flow characteristics and heat convection problems for elliptic ducts. [PERSON] [1] disclosed the experimental model of convection in elliptic conduits. [PERSON] et al. [2] reported the heat flux for a heated liquid in an elliptic domain geometry. [PERSON] et al. [3] presented the mathematical interpretation of non-Newtonian forced convection flow in an elliptic duct. Our main intention, in this work, is to analyse the convection phenomenon in an elliptic domain having heated deformable walls.  
Peristalsis mainly deals with the flow problems that involve the movement of channel walls to propagate the flow. Its comprehensive and broad range of practical applications in engineering problems make it a worthy topic of interest. [PERSON] and [PERSON] [4] disclosed the mathematical study of peristaltic flow and it is basically propagated in the tube due to deformation of the tube's walls. [PERSON] and [PERSON] [5] reported the mathematical analysis that deals with the non-Newtonian peristaltic flow problem. The peristaltic flow phenomenon, due to its practical field applications, is not just confined to cylindrical geometries. Many researchers reported recent research on peristaltic flow in view of multiple liquids and geometry domain models, such as non-uniform tubes [6, 7], diverging tube [8, 9], curved tube [10, 11], rectangular ducts [12, 13, 14] and elliptic duct [15].  
The metachronal wave effect on peristaltic flow is also mathematically investigated by many researchers. This metachronal wave is mainly generated due to rhythmic and sequential beating of cilia (hair-like tiny structures). The development of metachronal wave due to cilia also plays a key role in the flow propagation. [PERSON] and [PERSON] [16] disclosed the mathematical interpretation of the cilia-driven peristaltic flow of non-Newtonian fluids. [PERSON] et al. [17] scrutinized the transportation in a sinusoidal passage with cilia beating. [PERSON] et al. [18] disclosed the flow in a bent tube with effects of cilia beating and distinct physical properties of peristalsis. [PERSON] et al. [19] mathematically modelled the propulsion of [PERSON] non-Newtonian fluid in a sinusoidal pipe with the combined effects of cilia beating. Some recent studies on polymeric flow problems with weak form solution approaches are provided [20; 21; 22].  
The current assessment is done to mathematically interpret the peristaltic flow with convection for a non-Newtonian Jeffrey fluid in a ciliated elliptical conduit. The viscous dissipation effect is also integrated in the heat equation to provide a comprehensive interpretation of heat transfer for this convection heat transfer problem. A set of useful transformations and dimensionless parameters are employed to avail the dimensionless form of the mathematical problem. We developed exact results for pressure gradient, temperature and velocity. The distinct physical properties and heat convection aspects of this developed problem are analysed in detail through graphical outcomes. The increasing value of \(Q\) results in an enhanced contour size but a decline is observed in number of contours, as shown in streamline graphs.