In this paper, we are concerned with positive solutions for higher order m{point nonlinear fractional boundary value problems with integral boundary conditions. We establish the criteria for the existence of at least one, two and three positive solutions for higher order m{point nonlinear fractional boundary value problems with integral boundary conditions by using some results from the theory of  xed point index, Avery{Henderson  xed point theorem and the Legget{Williams  xed point theorem, respectively.

This article deals with a computational algorithmic approach for obtaining the optimal solution of a general free boundary problem governed by an elliptic equation with boundary control and functional criterion. After determining the weak solution of the system, the problem is converted into a variational format. Then, by using some aspects of measure theory, the method characterize the nearly optimal pair of domain and its related optimal control function at the same time. This method has many advantages such as strong linearity, automatic existence theorem and the ability of obtaining global solution. Two sets of numerical examples is also given.

In this note, we study Cauchy-Schwarz-type inequality for fractional Strum-Liouville boundary value problem containing Caputo derivative of order ɑ, , 1 < ɑ,≤,2. A lower bound for the smallest eigenvalue is determined using this inequality. We give a comparison between the smallest eigenvalue and its lower bound obtained from the Lyapunov-type and Cauchy-Schwarz-type inequalities which indi-cate the properties of eigenvalues.

A comparative study on numerical methods for Singularly Perturbed Boundary Turning Point Problems (SPBTPPs) featuring discontinuous source terms are examined. The study involves developing and analyzing two specific numerical techniques: the finite difference method and a hybrid difference method incorporating a Shishkin-type mesh. This approach demonstrates notable capabilities, exhibiting almost first-order and second-order convergence for the finite difference and hybrid difference methods, respectively. Numerical results are given  to support the theoretical findings.