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A study on single-iteration sobolev descent for linear initial value problems

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Abstract

Mahavier and Montgomery construct a Sobolev space for approximate solution of linear initial value problems in a finite difference setting in single-iteration sobolev descent for linear initial value problems, Mahavier, Montgomery, MJMS, 2013. Their Sobolev space is constructed so that gradient-descent converges to a solution in a single iteration, demonstrating the existence of a best Sobolev gradient for finite difference approximation of solutions of linear initial value problems. They then ask if there is a broader class of problems for which convergence in a single iteration in an appropriate Sobolev space occurs. We use their results to show the existence of single-step iteration to solution in a lower dimensional Sobolev space for their examples and then a class of problems for single-step convergence.

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Authors and Affiliations

  1. Department of Mathematics, Lahore University of Management Sciences, Lahore Cantt, Pakistan

    Sultan Sial & Adnan Khan

  2. Mathematics Department, Faculty of Science, Taibah University, Al-Madinah Al-Munawarah, Saudi Arabia

    Aly R. Seadawy

  3. Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore, Pakistan

    Nauman Raza & Ahmad Javid

Authors
  1. Sultan Sial
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  2. Aly R. Seadawy
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  3. Nauman Raza
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  4. Adnan Khan
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  5. Ahmad Javid
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Correspondence to Aly R. Seadawy.

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Sial, S., Seadawy, A.R., Raza, N. et al. A study on single-iteration sobolev descent for linear initial value problems. Opt Quant Electron 53, 135 (2021). https://doi.org/10.1007/s11082-021-02756-8

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