This paper presents a truly meshless approximation strategy for solving partial differential equations based on the local multiquadric (LMQ) and the local inverse multiquadric (LIMQ) approximations. It is different from the traditional global multiquadric (GMQ) approximation in such a way that it is a pure local procedure. In constructing the approximation function, the only geometrical data needed is the local configuration of nodes fallen within its influence domain. Besides this distinct characteristic of localization, in the context of meshlesstyped approximation strategies, other major advantages of the present strategy include: (i) the existence of the shape functions is guaranteed provided that all the nodal points within an influence domain are distinct; (ii) the constructed shape functions strictly satisfy the Kronecker delta condition; (iii) the approximation is stable and insensitive to the free parameter embedded in the formulation and; (iv) the computational cost is modest and the matrix operations require only inversion of matrices of small size which is equal to the number of nodes inside the influence domain. Based on the present LMQ and LIMQ approximations, a collocation procedure is developed for solutions of 1D and 2D boundary value problems. Numerical results indicate that the present LMQ and LIMQ approximations are more stable than their global counterparts. In addition, it demonstrates that both approximation strategies are highly efficient and able to yield accurate solutions regardless of the chosen value for the free parameter.Keywords Local multiquadric approximation, Local inverse multiquadric approximation, Radical base functions, Meshless method, Collocation procedure IntroductionRecently, the so-called ''meshless methods'' emerged as a potential alternative for solutions in computational mechanics and a variety of approaches named after ''meshless'' have appeared. They can be categorised into two main groups, with respect to their approximation techniques: (i) Methods based on the Galerkin integration technique For those methods grouped under this category, their final numerical equations are generated by substituting the approximation functions into an (Galerkin) integration equation. It includes the element-free Galerkin method [1, 2], the reproducing kernel particle method [3, 4], the h-p-cloud method [5], the partition-of-unity method [6], the meshless local Petrov-Galerkin method [7] and the point interpolation method [8]. (ii) Methods based on point collocation technique This group embraces methods which base on kinds of collocation technique to generate the final numerical equations. Examples include the finite point [9, 10] and the finite cloud [11] method, the multiquadric (MQ) and other methods in which the formulations are based on the radial basis functions (RBF) [12-18].In the above-mentioned meshless methods, the way of constructing their approximate field functions is one of the most important features that affect their performance, stability and efficiency. A wealth o...

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