In this paper, the necessary optimality conditions are studied and derived for a new class of the sum of two Caputo–Katugampola fractional derivatives of orders (α, ρ) and( β,ρ) with fixed the final boundary conditions. In the second study, the approximation of the left Caputo-Katugampola fractional derivative was obtained by using the shifted Chebyshev polynomials. We also use the Clenshaw and Curtis formula to approximate the integral from -1 to 1. Further, we find the critical points using the Rayleigh–Ritz method. The obtained approximation of the left fractional Caputo-Katugampola derivatives was added to the algorithm applied to the illustrative example so that we obtained the approximate results for the stat

In this paper, we present new algorithm for the solution of the nonlinear high order multi-point boundary value problem with suitable multi boundary conditions. The algorithm is based on the semi-analytic technique and the solutions are calculated in the form of a rapid convergent series. It is observed that the method gives more realistic series solution that converges very rapidly in physical problems. Illustrative examples are provided to demonstrate the efficiency and simplicity of the proposed method in solving this type of multi- point boundary value problems.

    In this paper, we present new algorithm for the solution of the second order nonlinear three-point boundary value problem with suitable multi boundary conditions. The algorithm is based on the semi-analytic technique and the solutions which are calculated in the form of a rapid convergent series. It is observed that the method gives more realistic series solution that converges very rapidly in physical problems. Illustrative examples are provided to demonstrate the efficiency and simplicity of the proposed method in solving this type of three point boundary value problems.

The aim of the research is to detect the relation between the fracture sets and systems with the stages of folding. The triple junction area of the research comprises the three faced plunges of three anticlines Bekhair, Brifca and Zawita anticline. GEOreint, ver 9.5.0 was used for analyzing and classifying the data collected from the field measurements on 11 stations in proportion to the orthogonal tectonic axes. The age of exposed rocks ranges from Paleocene up to Miocene. The fractures were represented as joints, veins in addition to different types of faults. The Kinematic analysis of the fractures revealed that the stress caused the (ac) and (hko> a) fractures is coincides with the regional compression stress that form the folds w

Oscillation criteria are obtained for all solutions of the first-order linear delay differential equations with positive and negative coefficients where we established some sufficient conditions so that every solution of (1.1) oscillate. This paper generalized the results in [11]. Some examples are considered to illustrate our main results.

Numerical Investigation was done for steady state laminar mixed convection and thermally and hydrodynamic fully developed flow through horizontal rectangular duct including circular core with two cases of time periodic boundary condition, first case  on the rectangular wall while keeping core wall constant and other on both the rectangular duct and core walls. The used governing equations are continuity momentum and energy equations. These equations are normalized and solved using the Vorticity-Stream function and the Body Fitted Coordinates (B.F.C.) methods. The Finite Difference approach with the Line Successive Over Relaxation (LSOR) method is used to obtain all the computational results the (B.F.C.) method is used to generate th

In this paper a modified approach have been used to find the approximate solution of ordinary delay differential equations with constant delay using the collocation method based on Bernstien polynomials.

Experimental investigations have been carried out to investigate the pH-control problems of industrial electroplating wastewater treatment plants. The accurate and sensitive PID control system could treat most problem and disturbances in the normal operation of the water treatment. However, conventional treatment was replaced by proprietary treatment agent called a QUASIL which was found to be more effective for a wide range of pH.

In this paper the modified trapezoidal rule is presented for solving Volterra linear Integral Equations (V.I.E) of the second kind and we noticed that this procedure is effective in solving the equations. Two examples are given with their comparison tables to answer the validity of the procedure.