Abstract
Problem statement: Here, we develop a discretized scheme using only the penalty method without involving the multiplier parameter to examine the convergence and geometric ratio profiles. Approach: This approach reduces computational time arising from less data manipulation. Objectively, we wish to obtain a numerical solution comparing favourably with the analytic solution.. Methodologically, we discretize the given problem, obtain an unconstrained formulation and construct an operator which sets the stage for the application of the discretized extended conjugate gradient method. Results: We analyse the efficiency of the developed scheme by considering an example and examining the generated sequential approximate solutions and the convergence ratio profile computed quadratically per cycle using the discretized conjugate gradient method. Conclusion/Recommendations: Both results, as shown in the table, look comparably and this suggests that the developed scheme may very well approximate an analytic solution of a given problem to an appreciable level of tolerance without its prior knowledge.
Highlights
Such that: Here, we examine a discretized scheme via the penalty method to examine the convergence analysis of optimal control problem constrained by evolution equation with real coefficients
We state the Theorem establishing operator V and provide a proof; Theorem 5.1 Let the initial guess of the conjugate gradient algorithm be z0(tk) so that z0(tk) = (x0(tk),u0(tk),h0(tk)), the control operator V associated with the generalized problem is given by Eq 11: Vz k 2
Considering the minimum of the objective functional values for each cycle, it is readily seen that the numerical solution approximates the analytic solution 0.2328 within an error tolerance of 0.021
Summary
Such that: Here, we examine a discretized scheme via the penalty method to examine the convergence analysis of optimal control problem constrained by evolution equation with real coefficients. Discretization of its time interval and finite difference method for its differential constraint were employed to obtain the discretized formulation of the problem. With this formulation, an associated operator was obtained using the modified extended Conjugate Gradient Method (CGM) (Hestenes, 1969; Ibiejugba and Onumanyi, 1984). Using the modified conjugate gradient method, an example was considered to examine the numerical solution and the convergence analysis as it compares favorably with each other To this end, a generalized quadratic problem constrained by evolution equation with real coefficients is considered in the paragraph for our developed scheme
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