Table 1.
Numerical results for Example 1
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Numerical investigation of space fractional order diffusion equation by the Chebyshev collocation method of the fourth kind and compact finite difference scheme
July
2021, 14(7): 2041-2053.
doi: 10.3934/dcdss.2020261
A new application of the reproducing kernel method
Siirt University, Art and Science Faculty Department of Mathematics 56100 Siirt, Turkey |
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.
Mathematics Subject Classification:
Primary: 47B32, 46E22, 39A10 Secondary: 65K15.
Citation:
Ali Akgül. A new application of the reproducing kernel method. Discrete and Continuous Dynamical Systems - S,
2021, 14
(7)
: 2041-2053.
doi: 10.3934/dcdss.2020261
![]()
References:
| [1] |
A. Akgül,
On the solution of higher-order difference equations, Mathematical Methods in the Applied Sciences, 40 (2017), 6165-6171.
doi: 10.1002/mma.3870. |
| [2] |
A. Akgül and E. Bonyah,
Reproducing kernel hilbert space method for the solutions of generalized kuramoto–sivashinsky equation, Journal of Taibah University for Science, 13 (2019), 661-669.
|
| [3] |
A. Akgül, M. Inc and E. Karatas,
Reproducing kernel functions for difference equations, Discrete & Continuous Dynamical Systems-Series S, 8 (2015), 1055-1064.
doi: 10.3934/dcdss.2015.8.1055. |
| [4] |
O. A. Arqub,
Computational algorithm for solving singular fredholm time-fractional partial integrodifferential equations with error estimates, Journal of Applied Mathematics and Computing, 59 (2019), 227-243.
doi: 10.1007/s12190-018-1176-x. |
| [5] |
F. V. Atkinson, Discrete and Continuous Boundary Problems, Mathematics in Science and Engineering, Vol. 8. Academic Press, New York, 1964. |
| [6] |
B. Azarnavid, M. Emamjome, M. Nabati and S. Abbasbandy, A reproducing kernel hilbert space approach in meshless collocation method, Computational and Applied Mathematics, 38 (2019), Art. 72, 19 pp.
doi: 10.1007/s40314-019-0838-0. |
| [7] |
M. Cui and Y. Lin, Nonlinear Numerical Analysis in the Reproducing Kernel Space, Nova Science Publishers Inc., New York, 2009. |
| [8] |
M. De la Sen,
The generalized beverton–holt equation and the control of populations, Applied Mathematical Modelling, 32 (2008), 2312-2328.
doi: 10.1016/j.apm.2007.09.007. |
| [9] |
M. Foroutan, R. Asadi and A. Ebadian, A reproducing kernel hilbert space method for solving the nonlinear three-point boundary value problems, International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, 32 (2019), e2573.
doi: 10.1002/jnm.2573. |
| [10] |
G. N. Gumah, M. F. M. Naser, M. Al-Smadi and S. K. Al-Omari, Application of reproducing kernel hilbert space method for solving second-order fuzzy volterra integro-differential equations, Advances in Difference Equations, 2018 (2018), Paper No. 475, 15 pp.
doi: 10.1186/s13662-018-1937-8. |
| [11] |
F. T. Isfahani and R. Mokhtari,
A numerical approach based on the reproducing kernel hilbert space for solving a class of boundary value optimal control problems, Iranian Journal of Science and Technology, Transactions A: Science, 42 (2018), 2309-2318.
doi: 10.1007/s40995-017-0421-8. |
| [12] |
B. S. H. Kashkari and M. I. Syam, Reproducing kernel method for solving nonlinear fractional fredholm integrodifferential equation, Complexity, 2018 (2018), 7pp.
doi: 10.1155/2018/2304858. |
| [13] |
W. G. Kelley and A. C. Peterson, Difference Equations, Academic Press Inc., Boston, MA, 1991. An introduction with applications. |
| [14] |
X. Li, H. Li and B. Wu,
Piecewise reproducing kernel method for linear impulsive delay differential equations with piecewise constant arguments, Applied Mathematics and Computation, 349 (2019), 304-313.
doi: 10.1016/j.amc.2018.12.054. |
| [15] |
S. H. Sababe and A. Ebadia,
Some properties of reproducing kernel banach and hilbert spaces, Sahand Communications in Mathematical Analysis, 12 (2018), 167-177.
|
| [16] |
Y.-L. Wang, Y. Liu, Z. Li and H. zhang,
Numerical solution of integro-differential equations of high-order fredholm by the simplified reproducing kernel method, International Journal of Computer Mathematics, 96 (2019), 585-593.
doi: 10.1080/00207160.2018.1455091. |
show all references
References:
| [1] |
A. Akgül,
On the solution of higher-order difference equations, Mathematical Methods in the Applied Sciences, 40 (2017), 6165-6171.
doi: 10.1002/mma.3870. |
| [2] |
A. Akgül and E. Bonyah,
Reproducing kernel hilbert space method for the solutions of generalized kuramoto–sivashinsky equation, Journal of Taibah University for Science, 13 (2019), 661-669.
|
| [3] |
A. Akgül, M. Inc and E. Karatas,
Reproducing kernel functions for difference equations, Discrete & Continuous Dynamical Systems-Series S, 8 (2015), 1055-1064.
doi: 10.3934/dcdss.2015.8.1055. |
| [4] |
O. A. Arqub,
Computational algorithm for solving singular fredholm time-fractional partial integrodifferential equations with error estimates, Journal of Applied Mathematics and Computing, 59 (2019), 227-243.
doi: 10.1007/s12190-018-1176-x. |
| [5] |
F. V. Atkinson, Discrete and Continuous Boundary Problems, Mathematics in Science and Engineering, Vol. 8. Academic Press, New York, 1964. |
| [6] |
B. Azarnavid, M. Emamjome, M. Nabati and S. Abbasbandy, A reproducing kernel hilbert space approach in meshless collocation method, Computational and Applied Mathematics, 38 (2019), Art. 72, 19 pp.
doi: 10.1007/s40314-019-0838-0. |
| [7] |
M. Cui and Y. Lin, Nonlinear Numerical Analysis in the Reproducing Kernel Space, Nova Science Publishers Inc., New York, 2009. |
| [8] |
M. De la Sen,
The generalized beverton–holt equation and the control of populations, Applied Mathematical Modelling, 32 (2008), 2312-2328.
doi: 10.1016/j.apm.2007.09.007. |
| [9] |
M. Foroutan, R. Asadi and A. Ebadian, A reproducing kernel hilbert space method for solving the nonlinear three-point boundary value problems, International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, 32 (2019), e2573.
doi: 10.1002/jnm.2573. |
| [10] |
G. N. Gumah, M. F. M. Naser, M. Al-Smadi and S. K. Al-Omari, Application of reproducing kernel hilbert space method for solving second-order fuzzy volterra integro-differential equations, Advances in Difference Equations, 2018 (2018), Paper No. 475, 15 pp.
doi: 10.1186/s13662-018-1937-8. |
| [11] |
F. T. Isfahani and R. Mokhtari,
A numerical approach based on the reproducing kernel hilbert space for solving a class of boundary value optimal control problems, Iranian Journal of Science and Technology, Transactions A: Science, 42 (2018), 2309-2318.
doi: 10.1007/s40995-017-0421-8. |
| [12] |
B. S. H. Kashkari and M. I. Syam, Reproducing kernel method for solving nonlinear fractional fredholm integrodifferential equation, Complexity, 2018 (2018), 7pp.
doi: 10.1155/2018/2304858. |
| [13] |
W. G. Kelley and A. C. Peterson, Difference Equations, Academic Press Inc., Boston, MA, 1991. An introduction with applications. |
| [14] |
X. Li, H. Li and B. Wu,
Piecewise reproducing kernel method for linear impulsive delay differential equations with piecewise constant arguments, Applied Mathematics and Computation, 349 (2019), 304-313.
doi: 10.1016/j.amc.2018.12.054. |
| [15] |
S. H. Sababe and A. Ebadia,
Some properties of reproducing kernel banach and hilbert spaces, Sahand Communications in Mathematical Analysis, 12 (2018), 167-177.
|
| [16] |
Y.-L. Wang, Y. Liu, Z. Li and H. zhang,
Numerical solution of integro-differential equations of high-order fredholm by the simplified reproducing kernel method, International Journal of Computer Mathematics, 96 (2019), 585-593.
doi: 10.1080/00207160.2018.1455091. |
| $x$ | 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 |
Table 2.
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 |
Table 3.
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}$ | $0.031$ |
Table 4.
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$ | $0.046$ |
| $80$ | $4.604129154\times 10^{118}$ | $4.604129154\times 10^{118}$ | $0.0$ | $0.0$ | $0.078$ |
| $90$ | $1.072315597\times 10^{138}$ | $1.072315597\times 10^{138}$ | $0.0$ | $0.0$ | $0.109$ |
| $100$ | $7.467889258\times 10^{157}$ | $7.467889258\times 10^{157}$ | $0.0$ | $0.0$ | $0.047$ |
Table 5.
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 |
Table 6.
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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Ali Akgül, Mustafa Inc, Esra Karatas. Reproducing kernel functions for difference equations. Discrete and Continuous Dynamical Systems - S, 2015, 8 (6) : 1055-1064. doi: 10.3934/dcdss.2015.8.1055 |
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Kaitlyn (Voccola) Muller. A reproducing kernel Hilbert space framework for inverse scattering problems within the Born approximation. Inverse Problems and Imaging, 2019, 13 (6) : 1327-1348. doi: 10.3934/ipi.2019058 |
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Irene Benedetti, Luisa Malaguti, Valentina Taddei. Nonlocal problems in Hilbert spaces. Conference Publications, 2015, 2015 (special) : 103-111. doi: 10.3934/proc.2015.0103 |
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