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This work is aim at providing a numerical technique for the Volterra integral equations using Galerkin method. For this purpose, an effective matrix formulation is proposed to solve linear Volterra integral equations of the first and second kind respectively using orthogonal polynomials as trial functions which are constructed in the interval [-1,1] with respect to the weight function w(x)=1+x
^{2}. The efficiency of the proposed method is tested on several numerical examples and compared with the analytic solutions available in the literature.

Modelling of physical problems arising from every discipline of study are often transformed into integral equations, namely, Volterra linear and nonlinear integral equations of the first and second kind respectively. For this, several authors have studied and applied these equations from the viewpoint of obtaining an analytic and unique numerical solution. In recent years, there has been a growing interest in the Volterra integral equations mainly due to its applicability in many areas of mathematical physics (astrophysics, contact problem, heat transfer problem and reactor theory). Consequently, most conventional analytic integral equations solvers have been developed and implemented since the digital computer was introduced some decades ago. The considerations are whether these solvers give an accurate solution, use less computation time, implement and give a compact solution form.

However, most of these solvers such as the Adomial decomposition method (ADM), Laplace transform method (LTM) and the Successive substitution method (SSM) do not have solutions in compact form. Thus, numerical stimulation in engineering science and in applied mathematics has become a powerful tool to model difficult phenomena, particularly, when analytic solutions are difficult to achieve.

Many researchers have developed numerical methods for the solution of Volterra integral equations using various polynomials. Rahman [

However, in this paper, an effective and efficient Galerkin numerical algorithm is formulated with orthogonal polynomials as basis which are constructed in the interval [−1, 1] with respect to the weight function w(x) = 1 + x^{2}. The proposed method is employed to solve linear Volterra integral of the first and second kind with regular and weak singular kernels, in details, in Section 3. Section 2 presents the concept of orthogonal polynomials. Section 4 presents numerical experiments of different kinds of Volterra integral equations to verify the proposed method. The results of each numerical example indicate convergence and error analysis are discussed. Finally, the conclusion is presented in Section 5.

Let

with the Kronecker delta

where the weight function w(x) is continuous and positive on [a, b] such that the moments

exist.

Then the integral,

denotes an inner product of polynomials

For orthogonality,

If

In this study, we adopt the weight function

The construction of

now follows:

Construction of Orthogonal Basis FunctionFor the purpose of constructing the basis function, we use additional property that

where

satisfies the orthogonality property (4).

Thus, the first six orthogonal polynomials

In this section, we first consider the Volterra integral of the second kind given by

where

Now, we use the Galerkin method to find an approximate solution

Now, substituting Equation (5) into Equation (7), we get

We obtain the Galerkin equation by multiplying both sides of Equation (8) by

Equation (9) is written in the matrix form as

where the elements of A,

Now, the unknown parameters are determined with a solver, which in this case is the Gaussian elimination method, and substituting these parameters in Equation (5), we get the approximate solution

Now, we consider the Volterra equation of the first kind given by

where

Applying the same procedure as described above, we obtain the matrix form

where the elements of A,

The unknown parameters are determined with a solver, which in this case is the Gaussian elimination method, and substituting these parameters in Equation (5), we get the approximate solution

The absolute error for this formulation is defined by absolute error

To illustrate the effectiveness of the proposed method, we demonstrate the method with five numerical examples which include first and second kind with regular and weakly kernels. For all examples considered, the solutions obtained by the proposed method are compared with the exact solutions available in the literature. The rate of convergence of each of the Linear Volterra integral equations is composed as

where

Example 1: Consider the linear Volterra integral equation of the first with continuous kernel [

The exact solution is

Example 2: Consider the first Abel’s linear Volterra integral equation [

The exact solution is

Example 3: Consider the second Abel’s linear Volterra integral equation of the form [

The exact solution is

Example 4: Consider the first Abel’s linear Volterra integral equation of the form [

where r is any positive number. The exact solution of the integral Equation (22) given by

In one numerical example r is chosen as

For

The maximum absolute errors obtain is in the order of

For

Example 5: Consider the second Abel’s linear Volterra integral equation of the form [

The exact solution is

In this paper, we have employed the Galerkin method based on the orthogonal polynomial basis tool which was constructed and has been developed to solve first and second kind Volterra integral equations. The numerical results obtained by the proposed method show an excellent rate of convergent even as n increases, which is shown in Tables 1-5. Also, the numerical solutions coincide with the exact solutions even at few numbers of

x | Exact Solutions | Approximate Solutions | Absolute Error |
---|---|---|---|

0.00 | 1.0000000 | 1.0372960 | 3.7296E−02 |

0.10 | 0.9910000 | 0.9907925 | 2.0746E−04 |

0.20 | 0.9680000 | 0.9519814 | 1.6019E−02 |

0.30 | 0.9370000 | 0.9208625 | 1.6138E−02 |

0.40 | 0.9040000 | 0.9208625 | 6.5641E−03 |

0.50 | 0.8750000 | 0.8974359 | 6.7016E−03 |

0.60 | 0.8560000 | 0.8817016 | 1.7660E−02 |

0.70 | 0.8530000 | 0.8736597 | 2.0310E−02 |

0.80 | 0.8720000 | 0.8733100 | 8.6527E−03 |

0.90 | 0.9190000 | 0.8956876 | 2.3312E−02 |

1.00 | 1.0000000 | 0.9184149 | 8.1585E−02 |

x | Exact Solutions | Approximate Solutions | Absolute Error |
---|---|---|---|

0.00 | 0.0000000 | 0.0854492 | 8.5449E−02 |

0.10 | 0.0000409 | 0.0056061 | 5.5652E−03 |

0.20 | 0.0009255 | −0.0361694 | 3.7095E−02 |

0.30 | 0.0057385 | −0.0398773 | 4.5616E−02 |

0.40 | 0.0209421 | −0.0055176 | 2.6460E−02 |

0.50 | 0.0571629 | 0.0669098 | 9.7468E−03 |

0.60 | 0.1298465 | 0.1774048 | 4.7558E−02 |

0.70 | 0.2598309 | 0.3259674 | 6.6137E−02 |

0.80 | 0.4738648 | 0.5125977 | 3.8733E−02 |

0.90 | 0.8050833 | 0.7372955 | 6.7788E−02 |

1.00 | 1.2934497 | 1.0000610 | 2.9339E−01 |

x | Exact Solutions | Approximate Solutions | Absolute Error |
---|---|---|---|

0.00 | 0.0000000 | −0.0213623 | 2.1362E−02 |

0.10 | 0.0000409 | 0.0062002 | 6.1593E−03 |

0.20 | 0.0009255 | 0.0097061 | 8.7806E−03 |

0.30 | 0.0057385 | 0.0063387 | 6.0023E−04 |

0.40 | 0.0209421 | 0.0132812 | 7.6608E−03 |

0.50 | 0.0571629 | 0.0477171 | 9.4458E−03 |

0.60 | 0.1298465 | 0.1268295 | 3.0169E−03 |

0.70 | 0.2598309 | 0.2678019 | 7.9710E−03 |

0.80 | 0.4738648 | 0.4878174 | 1.3953E−02 |

0.90 | 0.8050833 | 0.8040594 | 1.0239E−03 |

1.00 | 1.2934497 | 1.2337112 | 5.9738E−02 |

x | Exact Solutions | Approximate Solutions | Absolute Error |
---|---|---|---|

0.00 | 1.0000000 | 1.0628855 | 6.2886E−02 |

0.10 | 1.2156880 | 1.2243500 | 8.6620E−03 |

0.20 | 1.4656833 | 1.4456876 | 1.9996E−02 |

0.30 | 1.7548164 | 1.7268981 | 2.7918E−02 |

0.40 | 2.0885546 | 2.0679816 | 2.0573E−02 |

0.50 | 2.4730819 | 2.4689380 | 4.1439E−03 |

0.60 | 2.9153901 | 2.9297675 | 1.4377E−02 |

0.70 | 3.4233796 | 3.4504699 | 2.7090E−02 |

0.80 | 4.0059737 | 4.0310452 | 2.5072E−02 |

0.90 | 4.6732459 | 4.6714936 | 1.7523E−03 |

1.00 | 5.4365637 | 5.3718149 | 6.4749E−02 |

x | Exact Solutions | Approximate Solutions | Absolute Error |
---|---|---|---|

0.00 | 1.0000000 | 0.9945330 | 5.4670E−03 |

0.10 | 1.2156880 | 1.2167930 | 1.1050E−03 |

0.20 | 1.4656833 | 1.4679686 | 2.2853E−03 |

0.30 | 1.7548164 | 1.7556649 | 8.4845E−04 |

0.40 | 2.0885546 | 2.0874877 | 1.0669E−03 |

0.50 | 2.4730819 | 2.4710422 | 2.0398E−03 |

0.60 | 2.9153901 | 2.9139341 | 1.4560E−03 |

0.70 | 3.4233796 | 3.4237689 | 3.8930E−04 |

0.80 | 4.0059737 | 4.0081521 | 2.1784E−03 |

0.90 | 4.6732459 | 4.6746893 | 1.4434E−03 |

1.00 | 5.4365637 | 5.4309860 | 5.5777E−03 |

polynomials employed to find the approximate solution. Thus, the method is effective, efficient and reliable for the solution of other integral equations of other types.

James E.Mamadu,Ignatius N.Njoseh, (2016) Numerical Solutions of Volterra Equations Using Galerkin Method with Certain Orthogonal Polynomials. Journal of Applied Mathematics and Physics,04,367-382. doi: 10.4236/jamp.2016.42044