This paper is concerned with finding the approximation solution (APPS) of a certain type of nonlinear hyperbolic boundary value problem (NOLHYBVP). The given BVP is written in its discrete (DI) weak form (WEF), and is proved that it has a unique APPS, which is obtained via the mixed Galerkin finite element method (GFE) with implicit method (MGFEIM) that reduces the problem to solve the Galerkin nonlinear algebraic system (GNAS). In this part, the predictor and the corrector technique (PT and CT) are proved convergent and are used to transform the obtained GNAS to linear (GLAS ), then the GLAS is solved using the Cholesky method (ChMe). The stability and the convergence of the method are studied. The results are given by figures and shown the efficiency and accuracy for the method
In this paper, the classical continuous triple optimal control problem (CCTOCP) for the triple nonlinear parabolic boundary value problem (TNLPBVP) with state vector constraints (SVCs) is studied. The solvability theorem for the classical continuous triple optimal control vector CCTOCV with the SVCs is stated and proved. This is done under suitable conditions. The mathematical formulation of the adjoint triple boundary value problem (ATHBVP) associated with TNLPBVP is discovered. The Fréchet derivative of the Hamiltonian" is derived. Under suitable conditions, theorems of necessary and sufficient conditions for the optimality of the TNLPBVP with the SVCs are stated and proved.
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
... Show MoreIn this research, our aim is to study the optimal control problem (OCP) for triple nonlinear elliptic boundary value problem (TNLEBVP). The Mint-Browder theorem is used to prove the existence and uniqueness theorem of the solution of the state vector for fixed control vector. The existence theorem for the triple continuous classical optimal control vector (TCCOCV) related to the TNLEBVP is also proved. After studying the existence of a unique solution for the triple adjoint equations (TAEqs) related to the triple of the state equations, we derive The Fréchet derivative (FD) of the cost function using Hamiltonian function. Then the theorems of necessity conditions and the sufficient condition for optimality of
... Show MoreIn this paper, we have been used the Hermite interpolation method to solve second order regular boundary value problems for singular ordinary differential equations. The suggest method applied after divided the domain into many subdomains then used Hermite interpolation on each subdomain, the solution of the equation is equal to summation of the solution in each subdomain. Finally, we gave many examples to illustrate the suggested method and its efficiency.
In this research, we study the classical continuous Mixed optimal control vector problem dominated by couple nonlinear elliptic PDEs. The existence theorem for the unique state vector solution of the considered couple nonlinear elliptic PDEs for a given continuous classical mixed control vector is stated and proved by applying the Minty-Browder theorem under suitable conditions. Under suitable conditions, the existence theorem of a classical continuous mixed optimal control vector associated with the considered couple nonlinear elliptic PDEs is stated and proved.
In this paper we shall generalize fifth explicit Runge-Kutta Feldberg(ERKF(5)) and Continuous explicit Runge-Kutta (CERK) method using shooting method to solve second order boundary value problem which can be reduced to order one.These methods we shall call them as shooting Continuous Explicit Runge-Kutta method, the results are computed using matlab program.
In this article, the boundary value problem of convection propagation through the permeable fin in a natural convection environment is solved by the Haar wavelet collocation method (HWCM). We also compare the solutions with the application of a semi-analytical method , namely the Temimi and Ansari (TAM), that is characterized by accuracy and efficiency.The proposed method is also characterized by simplicity and efficiency. The possibility of applying the proposed method to many types of linear or nonlinear ordinary and partial differential equations.
In this paper, the methods of weighted residuals: Collocation Method (CM), Least Squares Method (LSM) and Galerkin Method (GM) are used to solve the thin film flow (TFF) equation. The weighted residual methods were implemented to get an approximate solution to the TFF equation. The accuracy of the obtained results is checked by calculating the maximum error remainder functions (MER). Moreover, the outcomes were examined in comparison with the 4th-order Runge-Kutta method (RK4) and good agreements have been achieved. All the evaluations have been successfully implemented by using the computer system Mathematica®10.
We study one example of hyperbolic problems it's Initial-boundary string problem with two ends. In fact we look for the solution in weak sense in some sobolev spaces. Also we use energy technic with Galerkin's method to study some properties for our problem as existence and uniqueness
In this paper, our purpose is to study the classical continuous optimal control (CCOC) for quaternary nonlinear parabolic boundary value problems (QNLPBVPs). The existence and uniqueness theorem (EUTh) for the quaternary state vector solution (QSVS) of the weak form (WF) for the QNLPBVPs with a given quaternary classical continuous control vector (QCCCV) is stated and proved via the Galerkin Method (GM) and the first compactness theorem under suitable assumptions(ASSUMS). Furthermore, the continuity operator for the existence theorem of a QCCCV dominated by the QNLPBVPs is stated and proved under suitable conditions.