# American Institute of Mathematical Sciences

## A new application of the reproducing kernel method

 Siirt University, Art and Science Faculty Department of Mathematics 56100 Siirt, Turkey

* Corresponding author: Ali Akgül

Received  April 2019 Revised  June 2019 Published  February 2020

We give a new implementation of the reproducing kernel method to investigate difference equations in this paper. We obtain the solutions in terms of convergent series. The method of obtaining the approximate solution in form of an algorithm is presented. We demonstrate some experiments to prove the accuracy of the technique.

Citation: Ali Akgül. A new application of the reproducing kernel method. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020261
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Numerical results for Example 1
 $x$ ES AS AE RE CPU time(s) 50 $1.125899907\times 10^{15}$ $1.125899907\times 10^{15}$ $0.0$ $0.0$ 0.078 100 $1.2676506\times 10^{30}$ $1.2676506\times 10^{30}$ $0.0$ $0.0$ 0.032 150 $1.427247693\times 10^{45}$ $1.42721\times 10^{45}$ $0.0$ $0.0000264095715$ 0.094 200 $1.606938044\times 10^{60}$ $1.606938044\times 10^{60}$ $0.0$ $0.0$ 0.031 250 $1.809251394\times 10^{75}$ $1.809251394\times 10^{75}$ $0.0$ $0.0$ 0.063 300 $2.037035976\times 10^{90}$ $2.037035976\times 10^{90}$ $0.0$ $0.0$ 0.078
 $x$ ES AS AE RE CPU time(s) 50 $1.125899907\times 10^{15}$ $1.125899907\times 10^{15}$ $0.0$ $0.0$ 0.078 100 $1.2676506\times 10^{30}$ $1.2676506\times 10^{30}$ $0.0$ $0.0$ 0.032 150 $1.427247693\times 10^{45}$ $1.42721\times 10^{45}$ $0.0$ $0.0000264095715$ 0.094 200 $1.606938044\times 10^{60}$ $1.606938044\times 10^{60}$ $0.0$ $0.0$ 0.031 250 $1.809251394\times 10^{75}$ $1.809251394\times 10^{75}$ $0.0$ $0.0$ 0.063 300 $2.037035976\times 10^{90}$ $2.037035976\times 10^{90}$ $0.0$ $0.0$ 0.078
Numerical results for Example 2
 $x$ ES AS AE RE CPU time(s) $10$ $39.88315482$ $39.88315374$ $1.08\times 10^{-6}$ $2.707910156\times 10^{−8}$ 0.344 $20$ $19.99998093$ $19.99998111$ $1.8\times 10^{-7}$ $9.000008582\times 10^{-9}$ 0.421 $30$ $39.99999989$ $39.99999960$ $2.9\times 10^{-7}$ $7.25000002\times 10^{-9}$ 0.344 $40$ $20.00000000$ $19.99999620$ $3.8\times 10^{-6}$ $1.9\times 10^{-7}$ 0.390 $50$ $40.00000000$ $39.99999340$ $6.6\times 10^{-6}$ $1.65\times 10^{-7}$ 0.437 $100$ $20.00000000$ $19.99999843$ $1.57\times 10^{-6}$ $7.85\times 10^{-8}$ 0.405
 $x$ ES AS AE RE CPU time(s) $10$ $39.88315482$ $39.88315374$ $1.08\times 10^{-6}$ $2.707910156\times 10^{−8}$ 0.344 $20$ $19.99998093$ $19.99998111$ $1.8\times 10^{-7}$ $9.000008582\times 10^{-9}$ 0.421 $30$ $39.99999989$ $39.99999960$ $2.9\times 10^{-7}$ $7.25000002\times 10^{-9}$ 0.344 $40$ $20.00000000$ $19.99999620$ $3.8\times 10^{-6}$ $1.9\times 10^{-7}$ 0.390 $50$ $40.00000000$ $39.99999340$ $6.6\times 10^{-6}$ $1.65\times 10^{-7}$ 0.437 $100$ $20.00000000$ $19.99999843$ $1.57\times 10^{-6}$ $7.85\times 10^{-8}$ 0.405
Numerical results for Example 3
 $x$ ES AS AE RE CPU $10$ $0.004173544379$ $0.004173091$ $4.53379\times 10^{-7}$ $0.000108631647$ $0.046$ $20$ $0.000004074670078$ $0.000003549$ $5.25670078\times 10^{-7}$ $0.1290092371$ $0.078$ $30$ $3.979168669\times 10^{-9}$ $6.56\times 10^{-7}$ $6.520208313\times 10^{-7}$ $163.8585558$ $0.063$ $40$ $3.885906902\times 10^{-12}$ $1.66\times 10^{-7}$ $1.659961141\times 10^{-7}$ $42717.47067$ $0.031$ $50$ $3.794830960\times 10^{-15}$ $3.68\times 10^{-7}$ $3.679999962\times 10^{-7}$ $9.697401546\times 10^{7}$ $0.032$ $60$ $3.705889610\times 10^{-18}$ $9.72\times 10^{-7}$ $9.720000000\times 10^{-7}$ $2.622852007\times 10^{11}$ $0.078$ $70$ $3.70588961\times 10^{-18}$ $6.891\times 10^{-7}$ $6.891000000\times 10^{-7}$ $1.859472549\times 10^{11}$ $0.078$ $80$ $3.53421174\times 10^{-24}$ $1.675\times 10^{-7}$ $1.675\times 10^{-7}$ $4.739387799\times 10^{16}$ $0.078$ $90$ $3.451378652\times 10^{-27}$ $7.3\times 10^{-8}$ $7.3\times 10^{-8}$ $2.115096817\times 10^{19}$ $0.047$ $100$ $3.370486965\times 10^{-30}$ $0.00000104$ $0.000001048$ $3.109342985\times 10^{23}$ $0.031$
 $x$ ES AS AE RE CPU $10$ $0.004173544379$ $0.004173091$ $4.53379\times 10^{-7}$ $0.000108631647$ $0.046$ $20$ $0.000004074670078$ $0.000003549$ $5.25670078\times 10^{-7}$ $0.1290092371$ $0.078$ $30$ $3.979168669\times 10^{-9}$ $6.56\times 10^{-7}$ $6.520208313\times 10^{-7}$ $163.8585558$ $0.063$ $40$ $3.885906902\times 10^{-12}$ $1.66\times 10^{-7}$ $1.659961141\times 10^{-7}$ $42717.47067$ $0.031$ $50$ $3.794830960\times 10^{-15}$ $3.68\times 10^{-7}$ $3.679999962\times 10^{-7}$ $9.697401546\times 10^{7}$ $0.032$ $60$ $3.705889610\times 10^{-18}$ $9.72\times 10^{-7}$ $9.720000000\times 10^{-7}$ $2.622852007\times 10^{11}$ $0.078$ $70$ $3.70588961\times 10^{-18}$ $6.891\times 10^{-7}$ $6.891000000\times 10^{-7}$ $1.859472549\times 10^{11}$ $0.078$ $80$ $3.53421174\times 10^{-24}$ $1.675\times 10^{-7}$ $1.675\times 10^{-7}$ $4.739387799\times 10^{16}$ $0.078$ $90$ $3.451378652\times 10^{-27}$ $7.3\times 10^{-8}$ $7.3\times 10^{-8}$ $2.115096817\times 10^{19}$ $0.047$ $100$ $3.370486965\times 10^{-30}$ $0.00000104$ $0.000001048$ $3.109342985\times 10^{23}$ $0.031$
Numerical results for Example 4
 $x$ ES AS AE RE CPU time(s) $10$ $1.543083500\times 10^{7}$ $1.543083500\times 10^{7}$ $0.0$ $0.0$ $0.063$ $20$ $1.890687562\times 10^{19}$ $1.890687562\times 10^{19}$ $0.0$ $0.0$ $0.140$ $30$ $2.996465452\times 10^{33}$ $2.996465452\times 10^{33}$ $0.0$ $0.0$ $0.094$ $40$ $1.209470191\times 10^{49}$ $1.209470191\times 10^{49}$ $0.0$ $0.0$ $0.031$ $50$ $1.240894842\times 10^{64}$ $1.240894842\times 10^{64}$ $0.0$ $0.0$ $0.016$ $60$ $4.047603113\times 10^{81}$ $4.047603113\times 10^{81}$ $0.0$ $0.0$ $0.125$ $70$ $6.766335157\times 10^{99}$ $6.766335157\times 10^{99}$ $0.0$ $0.0$ $0.046$ $80$ $4.604129154\times 10^{118}$ $4.604129154\times 10^{118}$ $0.0$ $0.0$ $0.078$ $90$ $1.072315597\times 10^{138}$ $1.072315597\times 10^{138}$ $0.0$ $0.0$ $0.109$ $100$ $7.467889258\times 10^{157}$ $7.467889258\times 10^{157}$ $0.0$ $0.0$ $0.047$
 $x$ ES AS AE RE CPU time(s) $10$ $1.543083500\times 10^{7}$ $1.543083500\times 10^{7}$ $0.0$ $0.0$ $0.063$ $20$ $1.890687562\times 10^{19}$ $1.890687562\times 10^{19}$ $0.0$ $0.0$ $0.140$ $30$ $2.996465452\times 10^{33}$ $2.996465452\times 10^{33}$ $0.0$ $0.0$ $0.094$ $40$ $1.209470191\times 10^{49}$ $1.209470191\times 10^{49}$ $0.0$ $0.0$ $0.031$ $50$ $1.240894842\times 10^{64}$ $1.240894842\times 10^{64}$ $0.0$ $0.0$ $0.016$ $60$ $4.047603113\times 10^{81}$ $4.047603113\times 10^{81}$ $0.0$ $0.0$ $0.125$ $70$ $6.766335157\times 10^{99}$ $6.766335157\times 10^{99}$ $0.0$ $0.0$ $0.046$ $80$ $4.604129154\times 10^{118}$ $4.604129154\times 10^{118}$ $0.0$ $0.0$ $0.078$ $90$ $1.072315597\times 10^{138}$ $1.072315597\times 10^{138}$ $0.0$ $0.0$ $0.109$ $100$ $7.467889258\times 10^{157}$ $7.467889258\times 10^{157}$ $0.0$ $0.0$ $0.047$
Numerical results for Example 5
 $x$ ES AS AE RE CPU time(s) 50 $1.258626873\times 10^{10}$ $1.258626873\times 10^{10}$ $0.0$ 0.0 2.465 100 $3.542248316\times 10^{20}$ $3.542248316\times 10^{20}$ $0.0$ 0.0 2.386 150 $9.969215980\times 10^{30}$ $9.969215980\times 10^{30}$ $0.0$ 0.0 2.512 200 $2.805711470\times 10^{41}$ $2.805711470\times 10^{41}$ $0.0$ 0.0 2.247 250 $7.896324908\times 10^{51}$ $7.896324908\times 10^{51}$ $0.0$ 0.0 2.340 300 $2.222322136\times 10^{62}$ $2.222322136\times 10^{62}$ $0.0$ 0.0 2.168 350 $6.254448414\times 10^{72}$ $6.254448414\times 10^{72}$ $0.0$ 0.0 2.247 400 $1.760236480\times 10^{83}$ $1.760236480\times 10^{83}$ $0.0$ 0.0 2.262
 $x$ ES AS AE RE CPU time(s) 50 $1.258626873\times 10^{10}$ $1.258626873\times 10^{10}$ $0.0$ 0.0 2.465 100 $3.542248316\times 10^{20}$ $3.542248316\times 10^{20}$ $0.0$ 0.0 2.386 150 $9.969215980\times 10^{30}$ $9.969215980\times 10^{30}$ $0.0$ 0.0 2.512 200 $2.805711470\times 10^{41}$ $2.805711470\times 10^{41}$ $0.0$ 0.0 2.247 250 $7.896324908\times 10^{51}$ $7.896324908\times 10^{51}$ $0.0$ 0.0 2.340 300 $2.222322136\times 10^{62}$ $2.222322136\times 10^{62}$ $0.0$ 0.0 2.168 350 $6.254448414\times 10^{72}$ $6.254448414\times 10^{72}$ $0.0$ 0.0 2.247 400 $1.760236480\times 10^{83}$ $1.760236480\times 10^{83}$ $0.0$ 0.0 2.262
Numerical results for Example 6
 $x$ ES AS AE CPU time(s) $10$ $-1$ $-0.9999995$ $5\times 10^{-7}$ $0.094$ $20$ $1$ $0.999902593$ $9.7407\times 10^{-5}$ $0.109$ $30$ $0$ $0.000006701$ $6.701\times 10^{-7}$ $0.110$ $40$ $-1$ $-0.999991867$ $8.133\times 10^{-6}$ $0.062$ $50$ $1$ $0.999997854$ $2.146\times 10^{-6}$ $0.063$ $60$ $0$ $0.000008068$ $8.068\times 10^{-6}$ $0.15$ $70$ $-1$ $-0.999983411$ $1.6589\times 10^{-5}$ $0.062$ $80$ $1$ $0.999969194$ $3.0806\times 10^{-5}$ $0.062$ $90$ $0$ $0.000008144$ $8.144\times 10^{-6}$ $0.78$ $100$ $-1$ $-0.999917201$ $8.2799\times 10^{-5}$ $0.047$
 $x$ ES AS AE CPU time(s) $10$ $-1$ $-0.9999995$ $5\times 10^{-7}$ $0.094$ $20$ $1$ $0.999902593$ $9.7407\times 10^{-5}$ $0.109$ $30$ $0$ $0.000006701$ $6.701\times 10^{-7}$ $0.110$ $40$ $-1$ $-0.999991867$ $8.133\times 10^{-6}$ $0.062$ $50$ $1$ $0.999997854$ $2.146\times 10^{-6}$ $0.063$ $60$ $0$ $0.000008068$ $8.068\times 10^{-6}$ $0.15$ $70$ $-1$ $-0.999983411$ $1.6589\times 10^{-5}$ $0.062$ $80$ $1$ $0.999969194$ $3.0806\times 10^{-5}$ $0.062$ $90$ $0$ $0.000008144$ $8.144\times 10^{-6}$ $0.78$ $100$ $-1$ $-0.999917201$ $8.2799\times 10^{-5}$ $0.047$
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