The approximate solution of a nonlinear parabolic boundary value problem with variable coefficients (NLPBVPVC) is found by using mixed Galekin finite element method (GFEM) in space variable with Crank Nicolson (C-N) scheme in time variable. The problem is reduced to solve a Galerkin nonlinear algebraic system (NLAS), which is solved by applying the predictor and the corrector method (PCM), which transforms the NLAS into a Galerkin linear algebraic system (LAS). This LAS is solved once using the Cholesky technique (CHT) as it appears in the MATLAB package and once again using the General Cholesky Reduction Order Technique (GCHROT), the GCHROT is employed here at first time to play an important role for saving a massive time. Illustrative

In this paper, the asymptotic behavior of all solutions of impulsive neutral differential equations with positive and negative coefficients and with impulsive integral term was investigated. Some sufficient conditions were obtained to ensure that all nonoscillatory solutions converge to zero. Illustrative examples were given for the main results.

This paper has the interest of finding the approximate solution (APPS) of a nonlinear variable coefficients hyperbolic boundary value problem (NOLVCHBVP).  The given boundary value problem is written in its discrete weak form (WEFM) and proved  have a unique solution, which is obtained via the mixed Galerkin finite element with implicit method that reduces the problem to solve the Galerkin nonlinear algebraic system  (GNAS). In this part, the predictor and the corrector techniques (PT and CT, respectively) are proved at first convergence and then are used to transform  the obtained GNAS to a linear GLAS . Then the GLAS is solved using the Cholesky method (ChMe). The stability and the convergence of the method are stud

A non-polynomial spline (NPS) is an approximation method that relies on the triangular and polynomial parts, so the method has infinite derivatives of the triangular part of the NPS to compensate for the loss of smoothness inherited by the polynomial. In this paper, we propose polynomial-free linear and quadratic spline types to solve fuzzy Volterra integral equations (FVIE) of the 2nd kind with the weakly singular kernel (FVIEWSK) and Abel's type kernel. The linear type algorithm gives four parameters to form a linear spline. In comparison, the quadratic type algorithm gives five parameters to create a quadratic spline, which is more of a credit for the exact solution. These algorithms process kernel singularities with a simple techniqu

In this work, we employ a new normalization Bernstein basis for solving linear Freadholm of fractional integro-differential equations  nonhomogeneous  of the second type (LFFIDE_s). We adopt Petrov-Galerkian method (PGM) to approximate solution of the (LFFIDE_s) via normalization Bernstein basis that yields linear system. Some examples are given and their results are shown in tables and figures, the Petrov-Galerkian method (PGM) is very effective and convenient and overcome the difficulty of traditional methods. We solve this problem (LFFIDE_s) by the assistance of Matlab10.   

     This paper deals with the numerical solution of the discrete classical optimal control problem (DCOCP) governing by linear hyperbolic boundary value problem (LHBVP). The method which is used here consists of: the GFEIM " the Galerkin finite element method in space variable with the implicit finite difference method in time variable" to find the solution of the discrete state equation (DSE) and the solution of its corresponding discrete adjoint equation, where a discrete classical control (DCC) is given.  The gradient projection method with either the Armijo method (GPARM) or with the optimal method (GPOSM) is used to solve the minimization problem which is obtained from the necessary conditi

In this article, the solvability of some proposal types of the multi-fractional integro-partial differential system has been discussed in details by using the concept of abstract Cauchy problem and certain semigroup operators and some necessary and sufficient conditions. 

  This paper studies the existence of  positive solutions for the following boundary value problem :-
 
 y(b) 0 α y(a) - β y(a) 0     bta             f(y) g(t) λy    
 
 
The solution procedure follows using the Fixed point theorem and obtains that this problem has at least one positive solution .Also,it determines (  ) Eigenvalue which would be needed to find the positive solution .