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Solution of Initial Value Problems Using an Operational Matrix

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Abstract

In this paper, a novel operational matrix method is introduced. This method is based on the frame of linear cardinal B-spline. We call this method as the frame operational matrix (FOM) method. First, we construct the operational matrix from the frame by using a collocation method. We develop the FOM method using this operational matrix. We apply this method to solve initial value problems both linear first and second order and nonlinear first order. Comparison in between our solution \((U_F)\), exact solution \((U_e)\), Haar solution \((U_H)\) and Runge Kutta solution \((U_R)\) also provided. Moreover, stability analysis is given which shows that the FOM method is of second order. Several numerical examples are provided to validate our theory.

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Acknowledgements

We are very much thankful to the referees who give their precious time to read this manuscript and confer their judgement to improve the quality of the article.

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Authors and Affiliations

  1. Department of Mathematics, Veer Surendra Sai University of Technology, Burla, Sambalpur, Odisha, 768018, India

    Kshama Sagar Sahu & Mahendra Kumar Jena

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  1. Kshama Sagar Sahu
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  2. Mahendra Kumar Jena
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Correspondence to Kshama Sagar Sahu.

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Appendix

Appendix

Frame matrix for \(J=0\left( M=3\right) \)

$$\begin{aligned} F_0= & {} \left( \begin{array}{ccc} 0.6667 &{} 2.0000 &{} 0.6667 \\ -0.9428 &{} 0.0000 &{} 0.9428 \\ -0.6667 &{} 2.0000 &{} -0.6667 \end{array}\right) .\\ P_1= & {} \left( \begin{array}{ccc} 0.5000 &{} 0.4714 &{} -0.2500 \\ -0.1473 &{} -0.0000 &{} -0.0295 \\ 0.0000 &{} 0.0589 &{} -0.0000 \end{array}\right) .\\ P_2= & {} \left( \begin{array}{ccc} 0.2963 &{} 0.3536 &{} -0.2130 \\ 0.3536 &{} 0.2130 &{} -0.0000 \\ -0.0231 &{} -0.1768 &{} 0.1898 \end{array}\right) . \end{aligned}$$

Frame matrix for \(J=1\left( M=7\right) \)

$$\begin{aligned} F_0= & {} \left( \begin{array}{ccccccc} 0.2857 &{} 0.8571 &{} 1.4286 &{} 2.0000 &{} 1.4286 &{} 0.8571 &{} 0.2857 \\ -0.4041 &{} -1.2122 &{} -0.8081 &{} 0 &{} 0.8081 &{} 1.2122 &{} 0.4041 \\ -0.8081 &{} -0.4041 &{} 1.2122 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} -1.2122 &{} 0.4041 &{} 0.8081 \\ -0.2857 &{} -0.8571 &{} 0.2857 &{} 2.0000 &{} 0.2857 &{} -0.8571 &{} -0.2857 \\ -0.5714 &{} 1.1429 &{} -0.5714 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} -0.5714 &{} 1.1429 &{} -0.5714 \end{array}\right) .\\ F_1= & {} \left( \begin{array}{ccccccc} 0.0102 &{} 0.0918 &{} 0.2551 &{} 0.5000 &{} 0.7449 &{} 0.9082 &{} 0.9898 \\ 0.0144 &{} 0.1299 &{} 0.2958 &{} 0.3536 &{} 0.2958 &{} 0.1299 &{} 0.0144 \\ 0.0577 &{} 0.3391 &{} 0.2237 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0.2237 &{} 0.3391 &{} 0.0577 \\ 0.0102 &{} 0.0918 &{} 0.1633&{} 0 &{} -0.1633&{} -0.0918&{}-0.0102 \\ 0.0408 &{} 0.1122&{} -0.1531&{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0.1531 &{} -0.1122&{}-0.0408 \end{array}\right) .\\ P_1= & {} \left( \begin{array}{ccccccc} 0.5000 &{} 0.4714 &{} 0.0034 &{} 0.4276 &{} -0.2500 &{} 0.0190 &{} -0.4190\\ -0.1726 &{} 0.0000 &{} -0.0405 &{} 0.0405 &{} -0.0042 &{} -0.0017 &{} -0.0017\\ -0.0905 &{} 0.1280 &{} -0.0310 &{} -0.0000 &{} 0.0905 &{} -0.0362 &{} 0.0000\\ -0.0905 &{} -0.1280 &{} 0.0000 &{} 0.0310 &{} 0.0905 &{} 0 &{} -0.0362\\ -0.0000 &{} 0.1094 &{} -0.0539 &{} -0.0539 &{} 0.0000 &{} 0.0167 &{} -0.0167\\ 0.0000 &{} -0.0000 &{} 0.0960 &{} 0.0000 &{} -0.0000 &{} -0.0643 &{} 0.0000\\ 0.0000 &{} 0.0000 &{} 0.0000 &{} 0.0960 &{} -0.0000 &{} 0 &{} 0.0643 \end{array} \right) .\\ {P}_2= & {} \left( \begin{array}{ccccccc} 0.2925 &{} -0.3536 &{} 0.0058 &{} -0.4300 &{} 0.2092 &{} -0.0082 &{} 0.3918\\ 0.3536 &{} -0.2092 &{} -0.0014 &{} -0.0014 &{} 0.0000 &{} -0.0221 &{} 0.0221\\ 0.2398 &{} 0.3391 &{} -0.2920 &{} -0.0000 &{} 0.2398 &{} -0.0122 &{} 0.0000\\ 0.2602 &{} -0.3680 &{} 0.0000 &{} -0.2920 &{} 0.2602 &{} 0 &{} 0.0122\\ -0.0043 &{} 0.1768 &{} -0.0250 &{} 0.0250 &{} -0.1709 &{} 0.0014 &{} 0.0014\\ 0.0036 &{} 0.0051 &{} 0.4510 &{} 0.0000 &{} 0.0036 &{} -0.5681 &{} 0.0000\\ 0.0036 &{} -0.0051 &{} 0.0000 &{} 0.5490 &{} 0.0036 &{} 0 &{} -0.5681 \end{array} \right) . \end{aligned}$$

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Sahu, K.S., Jena, M.K. Solution of Initial Value Problems Using an Operational Matrix. Int. J. Appl. Comput. Math 6, 61 (2020). https://doi.org/10.1007/s40819-020-00810-9

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