# System of 16 coupled Integro-differential equations

by Farhad.R   Last Updated August 14, 2019 09:20 AM - source

I have a question: How can i solve numerically a system of 16 coupled first-order integro-differential equations?

$$- a{{s'}_{11}} + b{{s'}_{33}} + c{{s'}_{44}} + Ng_1^2(\int_0^z {({s_{14}}(t) + {s_{23}}(t))} dt){{s'}_{41}} + Ng_1^2(\int_0^z {({s_{41}}(t) + {s_{32}}(t))} dt){{s'}_{14}} + Ng_2^2(\int_0^z {({s_{13}}(t) + {s_{24}}(t))} dt){{s'}_{31}} + Ng_2^2(\int_0^z {({s_{31}}(t) + {s_{42}}(t))} dt){{s'}_{13}} = - (Ng_1^2 + Ng_2^2){s_{11}} - Ng_2^2{s_{33}} - Ng_1^2{s_{44}}\\$$ $$- a{{s'}_{22}} + b{{s'}_{44}} + c{{s'}_{33}} + Ng_1^2(\int_0^z {({s_{14}}(t) + {s_{23}}(t))} dt){{s'}_{32}} + Ng_1^2(\int_0^z {({s_{41}}(t) + {s_{32}}(t))} dt){{s'}_{23}} + Ng_2^2(\int_0^z {({s_{13}}(t) + {s_{24}}(t))} dt){{s'}_{42}} + Ng_2^2(\int_0^z {({s_{31}}(t) + {s_{42}}(t))} dt){{s'}_{24}} = - (Ng_1^2 + Ng_2^2){s_{22}} - Ng_2^2{s_{44}} - Ng_1^2{s_{33}}\\$$ $$- (a + d){{s'}_{33}} - Ng_1^2(\int_0^z {({s_{14}}(t) + {s_{23}}(t))} dt){{s'}_{32}} - Ng_1^2(\int_0^z {({s_{41}}(t) + {s_{32}}(t))} dt){{s'}_{23}} - Ng_2^2(\int_0^z {({s_{13}}(t) + {s_{24}}(t))} dt){{s'}_{31}} - Ng_2^2(\int_0^z {({s_{31}}(t) + {s_{42}}(t))} dt){{s'}_{13}} = - (Ng_1^2 + Ng_2^2){s_{33}} + Ng_2^2{s_{11}} + Ng_1^2{s_{22}}\\$$ $$- (a + d){{s'}_{44}} - Ng_1^2(\int_0^z {({s_{14}}(t) + {s_{23}}(t))} dt){{s'}_{41}} - Ng_1^2(\int_0^z {({s_{41}}(t) + {s_{32}}(t))} dt){{s'}_{14}} - Ng_2^2(\int_0^z {({s_{13}}(t) + {s_{24}}(t))} dt){{s'}_{42}} - Ng_2^2(\int_0^z {({s_{31}}(t) + {s_{42}}(t))} dt){{s'}_{24}} = - (Ng_1^2 + Ng_2^2){s_{44}} + Ng_2^2{s_{22}} + Ng_1^2{s_{11}}\\$$ $${a_1}{{s'}_{41}} - Ng_1^2(\int_0^z {({s_{41}}(t) + {s_{32}}(t))} dt)({{s'}_{11}} - {{s'}_{44}}) - Ng_2^2(\int_0^z {({s_{31}}(t) + {s_{42}}(t))} dt)({{s'}_{21}} - {{s'}_{43}}) = (Ng_1^2 + Ng_2^2){s_{41}}\\$$ $${a_2}{{s'}_{14}} + Ng_1^2(\int_0^z {({s_{14}}(t) + {s_{23}}(t))} dt)({{s'}_{44}} - {{s'}_{11}}) + Ng_2^2(\int_0^z {({s_{13}}(t) + {s_{24}}(t))} dt)({{s'}_{34}} - {{s'}_{21}}) = - (Ng_1^2 + Ng_2^2){s_{14}}\\$$ $${b_1}{{s'}_{31}} - Ng_1^2(\int_0^z {({s_{41}}(t) + {s_{32}}(t))} dt)({{s'}_{21}} - {{s'}_{34}}) - Ng_2^2(\int_0^z {({s_{31}}(t) + {s_{42}}(t))} dt)({{s'}_{11}} - {{s'}_{33}}) = (Ng_1^2 + Ng_2^2){s_{31}}\\$$ $${b_2}{{s'}_{14}} + Ng_1^2(\int_0^z {({s_{14}}(t) + {s_{23}}(t))} dt)({{s'}_{43}} - {{s'}_{12}}) + Ng_2^2(\int_0^z {({s_{13}}(t) + {s_{24}}(t))} dt)({{s'}_{33}} - {{s'}_{11}}) = - (Ng_1^2 + Ng_2^2){s_{13}}\\$$ $${b_1}{{s'}_{42}} - Ng_1^2(\int_0^z {({s_{41}}(t) + {s_{32}}(t))} dt)({{s'}_{12}} - {{s'}_{43}}) - Ng_2^2(\int_0^z {({s_{31}}(t) + {s_{42}}(t))} dt)({{s'}_{22}} - {{s'}_{44}}) = (Ng_1^2 + Ng_2^2){s_{42}}\\$$ $${b_2}{{s'}_{24}} + Ng_1^2(\int_0^z {({s_{14}}(t) + {s_{23}}(t))} dt)({{s'}_{34}} - {{s'}_{12}}) + Ng_2^2(\int_0^z {({s_{13}}(t) + {s_{24}}(t))} dt)({{s'}_{44}} - {{s'}_{22}}) = - (Ng_1^2 + Ng_2^2){s_{24}}\\$$ $${c_1}{{s'}_{32}} - Ng_1^2(\int_0^z {({s_{41}}(t) + {s_{32}}(t))} dt)({{s'}_{22}} - {{s'}_{33}}) - Ng_2^2(\int_0^z {({s_{31}}(t) + {s_{42}}(t))} dt)({{s'}_{21}} - {{s'}_{34}}) = (Ng_1^2 + Ng_2^2){s_{32}}\\$$ $${c_2}{{s'}_{24}} + Ng_1^2(\int_0^z {({s_{14}}(t) + {s_{23}}(t))} dt)({{s'}_{33}} - {{s'}_{22}}) + Ng_2^2(\int_0^z {({s_{13}}(t) + {s_{24}}(t))} dt)({{s'}_{43}} - {{s'}_{21}}) = - (Ng_1^2 + Ng_2^2){s_{23}}\\$$ $$- (a + d){{s'}_{43}} - Ng_1^2(\int_0^z {({s_{14}}(t) + {s_{23}}(t))} dt){{s'}_{42}} - Ng_1^2(\int_0^z {({s_{41}}(t) + {s_{32}}(t))} dt){{s'}_{13}} - Ng_2^2(\int_0^z {({s_{13}}(t) + {s_{24}}(t))} dt){{s'}_{41}} - Ng_2^2(\int_0^z {({s_{31}}(t) + {s_{42}}(t))} dt){{s'}_{23}} = (Ng_1^2 + Ng_2^2){s_{43}} + Ng_2^2{s_{21}} + Ng_1^2{s_{12}}\\$$ $$- (a + d){{s'}_{34}} - Ng_1^2(\int_0^z {({s_{14}}(t) + {s_{23}}(t))} dt){{s'}_{31}} - Ng_1^2(\int_0^z {({s_{41}}(t) + {s_{32}}(t))} dt){{s'}_{24}} - Ng_2^2(\int_0^z {({s_{13}}(t) + {s_{24}}(t))} dt){{s'}_{32}} - Ng_2^2(\int_0^z {({s_{31}}(t) + {s_{42}}(t))} dt){{s'}_{14}} = (Ng_1^2 + Ng_2^2){s_{34}} + Ng_2^2{s_{12}} + Ng_1^2{s_{21}}\\$$ $$- a{{s'}_{21}} + Ng_1^2(\int_0^z {({s_{14}}(t) + {s_{23}}(t))} dt){{s'}_{31}} + Ng_1^2(\int_0^z {({s_{41}}(t) + {s_{32}}(t))} dt){{s'}_{24}} + Ng_2^2(\int_0^z {({s_{13}}(t) + {s_{24}}(t))} dt){{s'}_{41}} + Ng_2^2(\int_0^z {({s_{31}}(t) + {s_{42}}(t))} dt){{s'}_{23}} = - (Ng_1^2 + Ng_2^2){s_{21}} - (Ng_1^2 + Ng_2^2){s_{34}}\\$$ $$- a{{s'}_{12}} + Ng_1^2(\int_0^z {({s_{14}}(t) + {s_{23}}(t))} dt){{s'}_{42}} + Ng_1^2(\int_0^z {({s_{41}}(t) + {s_{32}}(t))} dt){{s'}_{31}} + Ng_2^2(\int_0^z {({s_{13}}(t) + {s_{24}}(t))} dt){{s'}_{32}} + Ng_2^2(\int_0^z {({s_{31}}(t) + {s_{42}}(t))} dt){{s'}_{14}} = - (Ng_1^2 + Ng_2^2){s_{12}} - Ng_1^2{s_{43}} + Ng_2^2{s_{34}}\\$$

we have 16 variable $${s_{ij}}$$ and they depend on z, we are looking for them.N, a,b,c,d,$${g_1}$$,$${g_2}$$,$${a_1}$$,$${a_2}$$,$${b_1}$$,$${b_2}$$,$${c_1}$$ and $${c_2}$$ are constant.

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