# Comparing dierent degrees of nonlinearity for inverse problem for parabolic equation

## Keywords:

optimization, control, nonlinear parabolic equation, Gateaux derivative, approximation, gradient,## Abstract

In this work we consider one dimensional nonlinear parabolic equation with unknown function on the right side of space variable. As an additional information we are given a function which describes a solution on the left side and thus the problem is overdened on the left side. The problem is solved by gradient method. The main target is to understand an inuence of the nonlinearity degree of the equation on convergence of the numerical algorithm. For that we take dierent degrees of nonlinear term in the equation, construct a numerical solution and give the results in graphical form. Also we enlarge a time interval and consider a convergence of the algorithm. Some negative eects can be avoided by enlarging the time interval. We give all formulae to solve a direct problem and adjoint problem, give references where to nd how to obtain a gradient for the functional given on nonlinear parabolic equation. We also describe the step-by-step algorithm of the solution of the problem. Higher degrees of the nonlinearity make the numerical solution less accurate, but at the same time it makes the functional properties of the equation much better. Inuence of these two aspects is considered in the work. Also some comments are given on some moments for the numerical algorithm, such as choosing a constant coecient in gradient method.## References

[1] Becker J., Dagum L. Particle simulation on heterogeneous distributed supercomputers // Concurrency-Practice and experience. 1993. 5(4).-C. 367377.

[2] Fougere D., Malyshkin V. NumGrid middleware: MPI support for computational grids // Lecture Notes in Computer Science (PACT 2005). 2005. V.3606. P. 313320.

[3] Diaz J., Munoz-Caro C., Nino A.A. Survey of Parallel Programming Models and Tools in the Multi and Many-Core Era // IEEE Transactions on parallel and distributed systems. 2012. V.23(8). P. 13691386.

[4] Cappello F., Djilali S., Fedak G. Computing on large-scale distributed systems: XtremWeb architecture, programming models, security, tests and convergence with grid // Future generation computer systems. 2005. V. 21(3). P. 417437.

[5] Wang J., Liu Z. Parallel Data Mining Optimal Algorithm of Virtual Cluster // International Conference on Fuzzy Systems and Knowledge. 2008. V.5. P. 358362.

[6] Pandey S., Buyya R. Scheduling Workow Applications Based on Multi-source Parallel Data Retrieval in Distributed Computing Networks // Computer journal. 2012. V. 55(11). P. 12881308.

[7] Liu H., Orban D. GridBatch: Cloud Computing for Large-Scale Data-Intensive Batch Applications // CCGRID. 2008. V. 1. Ð.295305.

[8] Valilai O., Houshmand M. A collaborative and integrated platform to support distributed manufacturing system using a service-oriented approach based on cloud computing paradigm // Robotics and computer-integrated manufacturing. 2013. V.29(1). Ð.110127.

[9] Ahmed-Zaki D., Dobrowolski G., Kumalakov B. Peer-to-Peer MapReduce Platform // Proceedings of the 5th International Conference on Agents and Arti?cial Intelligence. (ICAART 2013). 2013. V. 2. P. 565570.

[10] Akhmed-Zaki D., Kumalakov B. Composite Peer-to-Peer MapReduce System // Proceedings of the International Conference on New Trends in Information and Communication Technologies (ICTT 2013). 2013. P.4449.

[11] Akhmed-Zaki D.Zh., Kumalakov B.A. Solving complex iterative tasks using intellectual load distribution and MPI // Вестник Национальной инженерной академии Республики Казахстан. 2014. 2(52). P. 2531.

[2] Fougere D., Malyshkin V. NumGrid middleware: MPI support for computational grids // Lecture Notes in Computer Science (PACT 2005). 2005. V.3606. P. 313320.

[3] Diaz J., Munoz-Caro C., Nino A.A. Survey of Parallel Programming Models and Tools in the Multi and Many-Core Era // IEEE Transactions on parallel and distributed systems. 2012. V.23(8). P. 13691386.

[4] Cappello F., Djilali S., Fedak G. Computing on large-scale distributed systems: XtremWeb architecture, programming models, security, tests and convergence with grid // Future generation computer systems. 2005. V. 21(3). P. 417437.

[5] Wang J., Liu Z. Parallel Data Mining Optimal Algorithm of Virtual Cluster // International Conference on Fuzzy Systems and Knowledge. 2008. V.5. P. 358362.

[6] Pandey S., Buyya R. Scheduling Workow Applications Based on Multi-source Parallel Data Retrieval in Distributed Computing Networks // Computer journal. 2012. V. 55(11). P. 12881308.

[7] Liu H., Orban D. GridBatch: Cloud Computing for Large-Scale Data-Intensive Batch Applications // CCGRID. 2008. V. 1. Ð.295305.

[8] Valilai O., Houshmand M. A collaborative and integrated platform to support distributed manufacturing system using a service-oriented approach based on cloud computing paradigm // Robotics and computer-integrated manufacturing. 2013. V.29(1). Ð.110127.

[9] Ahmed-Zaki D., Dobrowolski G., Kumalakov B. Peer-to-Peer MapReduce Platform // Proceedings of the 5th International Conference on Agents and Arti?cial Intelligence. (ICAART 2013). 2013. V. 2. P. 565570.

[10] Akhmed-Zaki D., Kumalakov B. Composite Peer-to-Peer MapReduce System // Proceedings of the International Conference on New Trends in Information and Communication Technologies (ICTT 2013). 2013. P.4449.

[11] Akhmed-Zaki D.Zh., Kumalakov B.A. Solving complex iterative tasks using intellectual load distribution and MPI // Вестник Национальной инженерной академии Республики Казахстан. 2014. 2(52). P. 2531.

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## How to Cite

*Journal of Mathematics, Mechanics and Computer Science*,

*83*(4), 3–11. Retrieved from https://bm.kaznu.kz/index.php/kaznu/article/view/74

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## Section

Mechanics, Mathematics, Computer Science